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Full text of "An introduction to astronomy : designed as a text book for the students of Yale College"



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Entered, according to Act of Congress, in the year 1839, 

In the Clerk's Office of the District Court of Connecticut. 

Stereotyped by 

45 Gold-street, New York. 


NEARLY all who have written Treatises on Astronomy, designed for young 
learners, appear to have erred in one of two ways ; they have either disre- 
garded demonstrative evidence, and relied on mere popular illustration, or they 
have exhibited the elements of the science in naked mathematical formula?. 
The former are usually diffuse and superficial ; the latter, technical and ab- 

In the following Treatise, we have endeavored to unite the advantages of 
both methods. We have sought, first, to establish the great principles of 
astronomy on a mathematical basis ; and, secondly, to render the study inter- 
esting and intelligible to the learner, by easy and familiar illustrations. We 
would not encourage any one to believe that he can enjoy a full view of the 
grand edifice of astronomy, while its noble foundations are hidden from his 
sight; nor would we assure him that he can contemplate the structure in its 
true magnificence, while its basement alone is within his field of vision. We 
would, therefore, that the student of astronomy should confine his attention 
neither to the exterior of the building, nor to the mere analytic investigation 
of its structure. We would desire that he should not only study it in models 
and diagrams, and mathematical formulas, but should at the same time acquire 
a love of nature herself, and cultivate the habit of raising his views to the 
grand originals. Nor is the effort to form a clear conception of the motions and 
dimensions of the heavenly bodies, less favorable to the improvement of the 
intellectual powers, than the study of pure geometry. 

But it is evidently possible to follow out all the intricacies of an analytical 
process, and to arrive at a full conviction of the great truths of astronomy, and 
yet know very little of nature. According to our experience, however, but few 
students in the course of a liberal education will feel satisfied with this. They 
do not need so much to be convinced that the assertions of astronomers are 
true, as they desire to know what the truths are, and how they were ascer- 
tained ; and they will derive from the study of astronomy little of that moral 
and intellectual elevation which they had anticipated, unless they learn to look 
upon the heavens with new views, and a clear comprehension of their won- 
derful mechanism. 

Much of the difficulty that usually attends the early progress of the astro- 
nomical student, arises from his being too soon introduced to the most perplex- 
ing part of the whole subject, the planetary motions. In this work, the con- 
sideration of these is for the most part postponed until the learner has become 
familiar with the artificial circles of the sphere, and conversant with the celes- 
tial bodies. We then first take the most simple view possible of the planetary 
motions by contemplating them as they really are in nature, and afterwards 
proceed to the more difficult inquiry, why they appear as they do. Probably 
no science derives such signal advantage from a happy arrangement, as as- 
tronomy ; an order, which brings out every fact or doctrine of the science just 
in the p'lace where the mind of the learner is prepared to receive it. 

Although we have found it convenient to defer the consideration of the fixed 
stars to a late period, yet we would earnestly recommend to the student to be- 
gin to learn the constellations, and the stars of the first magnitude at least, as 


soon as he enters upon the study of astronomy. A few evenings spent in this 
way, assisted, where it is practicable, by a friend already conversant with the 
stars, will inspire a higher degree of enthusiasm for the science, and render its 
explanations more easily understood. 

It is recommended to the learner to make a free use of the Analysis, espe- 
cially in reviewing the ground already traversed. If by repeated recurrence to 
these heads, he associates with each a train of ideas, carrying along with him, 
as he advances, all the particulars indicated in these hints, he will secure to 
them an indelible place in his memory. 

With such aids at hand, as Newton, La Place, and Delambre, to expound 
the laws of astronomy, and such popular writers as Ferguson, Biot, and Fran- 
cceur, to supply familiar illustrations of those laws, it might seem an easy task 
to prepare a work like the present ; but a text book made up of extracts from 
these authors, would be ill suited to the wants of our students. We have 
deemed it better therefore, first, to acquire the clearest views we were able of 
the truths to be unfolded, both from an extensive perusal of standard authors, 
and from diligent reflection, and then to endeavor to transfuse our own im- 
pressions into the mind of the learner. Writers of profound attainments in 
astronomy, and of the highest reputation, have often failed in the preparation 
of elementary works, because they lacked one qualification the experience of 
the teacher. Familiar as they were with the truths of the science, but unac- 
customed to hold communion with young pupils, they were incapable of ap- 
prehending the difficulty and the slowness with which these truths make their 
entrance into the mind for the first time. Even when they attempt to feel 
their way into young minds, by assuming the garb of the instructor, and em- 
ploying popular illustrations, they often betray their want of the experience 
and art of the professional teacher. 

Astronomy, in its grandest and noblest conceptions, addresses itself alike to 
the intellect and to the heart. It demands the highest efforts of the one and the 
warmest and most devout affections of the other, in order fully to comprehend 
its truths and to relish its sublimity. The task of learning the bare elements 
of this, as well as of every other science, is purely intellectual, and is to be re- 
garded only as preparing the way for that more enlarged and exalted contem- 
plation of the heavenly bodies, to which the mind will naturally rise, when it 
can view all things in their true relations to each other. It is therefore essen- 
tial to this study, as a part of a public education, that the student, after ac- 
quiring a knowledge of the elements of the science, should return to the sub- 
ject, and trace the great discoveries of astronomy, as they have succeeded one 
another from the earliest ages of society down to the present time, viewing them 
in connection with the many interesting historical and biographical incidents 
which attended their development. The author is therefore accustomed, in 
his own course of instruction, to follow the study of this "Introduction," with 
a course of Lectures adapted to such a purpose ; and, with similar views, he 
has prepared a volume of " Letters on Astronomy," where he has attempted 
to connect with the leading truths of the science such historical incidents and 
moral reflections, as may at once interest the understanding and amend the 




Astronomy defined, 

Descriptive Astronomy, 

Physical do. 

Practical do. 

History. Ancient nations who cultiva- 
ted astronomy, 

Pythagoras his age and country, 1 

His views of the celestial motions, 

Alexandrian School when founded 
by whom introduction of astronomi- 
cal instruments, 2 

Hipparchus his character, 2 

Ptolemy the Almagest, 2 

Copernicus, Tycho Brahe, Kepler and 
Galileo respective labors of each,.... 2 

Sir Isaac Newton his great discovery, 2 

La Place Mecanique Celeste, 2 

Astrology Natural and Judicial ob- 
ject of each, 2 

Accuracy aimed at by astronomers, 

Copernican System its leading doc- 
trines, 3 

Plan of this work, 3 




Figure of the earth, 4 

Proofs, 4 

Dip of the horizon, 4 

How found, 5 

Table of the dip its use, 6 

Exact figure of the earth, 6 

Its circumference, 6 

Small inequalities of the earth's surface, 6 

Diameter of the earth how determined, 7 
How to divest the mind of preconceived 

erroneous notions, 8 


Great and small circles defined, 9 

Axis of a circle pole, 9 

Situation of the poles of two great cir- 
cles which cut each other at right an- 

gles, 9 

Points of intersection of two great cir- 
cleshow many degrees apart, 


Page. When a great circle passes through the 

1 pole of another, how does it cut it ? . 10 

1 Secondary defined, 10"""" 

1 Angle made by two great circles how 

1 measured, 10 

Terrestrial and Celestial spheres distin- 

1 guished, 10 

Horizon defined, 11 

Sensible horizon, 11 

Rational do 11 

Zenith and Nadir, 11 

Vertical circles, 11 

Meridian, 11 

Prime Vertical, 11 

How the place of a celestial body is de- 
termined, 11 

Altitude azimuth amplitude, 12 

Zenith Distance how measured, 12 

Axis of the earth axis of the celestial 

sphere, 12 

Poles of the earth poles of the heav- 
ens, 12 

Equator terrestrial and celestial, 12 

Hour circles, 

Latitude, 13 

Polar Distance, how related to latitude, 13 

Longitude, 13 

OF THE Standard Meridians, 13 

Ecliptic, 13 

Inclination of the ecliptic to the equa- 
tor, 13 

Equinoctial points, 13 

Equinoxes Vernal and autumnal, .... 13 

Solstitial points, 14 

Solstices, 14 

Signs of the ecliptic enumerated, 14 

3olures Equinoctial and Solstitial,... 14 

Right ascension, 15 

Declination, 15 

elestial Longitude, 15 

Celestial Latitude, 15 

North Polar Distances, how related to 

latitude, 15 

Parallels of Latitude, 15 

Tropics, 16 

Polar circles, , 16 





10 Zodiac,.. 


Elevation of the pole to what is it 

equal? 16 

Elevation of the equator, 16 

Distance of a place from the pole, to 

what equal? 16 


Circles of Diurnal Revolution, 17 

Sidereal day defined, 17 

Appearance of the circles of diurnal 

revolution at the equator, 17 

ARight Sphere defined, 18 

A Parallel Sphere, 19 

An Oblique Sphere, 19 

Circle of Perpetual Apparition, 20 

Circle of Perpetual Occultation, 20 

How are the circles of daily motion cut 
by the horizon in the different 

spheres? 20 

Explanation of the peculiar appearan- 
ces of each sphere, from the revolu- 
tion of the earth on its axis, 21 

tificial Globes terrestrial and celes- 
tial, 22 

Their use, 23 

Meridian how represented how gra- 
duated, 23 

Horizon how represented how gra- 
duated, 23 

Hour Circles, how represented, 23 

Hour Index described, 23 

Quadrant of Altitude, 24 

Its use described, 24 

To rectify the globe for any place, 24 

To find the latitude and longitude 

of a place, 24 

To find a place, its latitude and longi- 

gitude being given, 25 

To find the bearing and distance of 

two places, 25 

To determine the difference of time of 

two places, 25 

The hour being given at any place, to 
tell what hour it is in any other part 

oftheworld, 25 

To find the antceci, periaeci, and antipo- 
des, 25 

To rectify the globe for the sun's place, 26 
The latitude of the place being given, 
to find the time of the sun's rising 

and setting, 26 

To find the right ascension and decli- 
nation, . 26 

To represent the appearance of the 

heavens at any time, 2' 

To find the altitude and azimuth of a 
star, 2 


To find the angular distance of two 
stars from each other, 27 

To find the surfs meridian altitude, 
the latitude and day of the month 
being given, 28 


*arallax defined, 28 

lorizontal Parallax, 29 

lelation of parallax to the zenith dis- 
tance, and distance from the center 

of the earth, 29 

To find the horizontal parallax from 

the parallax at any altitude, 29 

Amount of parallax in the zenith and 

in the horizon, 30 

fFect of parallax upon the altitude of 

a body, , 30 

Mode of determining the horizontal 

parallax of a body, 30 

Amount of the sun's hor. par 31 

Jse of parallax, 31 

Refraction. Its effect upon the alti- 
tude of a body, 32 

[ts nature illustrated, 32 

[ts amount at different angles of eleva- 
tion, 32 

How the amount is ascertained, 33 

Sources of inaccuracy in estimating the 

refraction, 35 

Effect of refraction upon the sun and 

moon when near the horizon, 35 

Oval figure of these bodies explained,. 35 
Apparent enlargement of the sun and 

moon near the horizon, 36 

Twilight. Its cause explained, 37 

Length of twilight in different latitudes, 37 
How the atmosphere contributes to dif- 
fuse the sun's light, 37 

Chapter IV. TIME. 

Time defined, 38 

What period is a sidereal day, 38 

Uniformity of sidereal days, 38 

Solar time, how reckoned, 

Why solar days are longer than side- 



Apparent time defined, 39 

Mean time, 40 

An astronomical day, 40 

Equation of time defined, 40 

When do apparent time and mean 

time differ most? 40 

When do they come together? 40 

Effect of a change in the place of the 

earth's perihelion, 40 

Causes of the inequality of the solar 

days, 41 

Explain the first cause, depending on 

the unequal velocities of the sun,.... 41 



Explain the second cause, depending 

on the obliquity of the ecliptic, 42 

When does the sidereal day com- 

mence? 44 

The Calendar. Astronomical year de- 
fined, 45 

How the most ancient nations deter- 

mined the number of days in the year, 45 
Julius Caesar's reformation of the calen. 

dar explained, 45 

Errors of this calendar, 45 

Reformation by Pope Gregory 46 

Rule for the Gregorian calendar, 46 

New style, when adopted in England, 46 
What nations still adhere to the old 

style? 46 

What number of days is now allowed 

between old and new style ? 47 

How the common year begins and ends, 47 

How leap year begins and ends, 47 

Does the confusion of different calen- 
dars affect astronomical observations ? 47 



How the most ancient nations acquired 

their knowledge of Astronomy, 48 

Use of instruments in the Alexandrian 

School, 48 

Ditto, by Tycho Brahe, 48 

Ditto, by the Astronomers Royal, 48 

Space occupied by l"on the limb of an 

instrument, 48 

Extent of actual divisions on the limb, 49 

Vernier, defined, 49 

Its use illustrated, 49 

Chief astronomical instruments enu- 
merated, 50 

Observations taken on the meridian. . . 50 

Reasons of this, 50 

Transit Instrument defined, 51 

Ditto described, 51 

Method of placing it in the meridian. . 51 

Line of collimation defined, 52 

System of wires in the focus, 52 

Its use for arcs of right ascension, 52 

Astronomical Clock, how regulated, 52 

What does it show ? 

How to test its accuracy, 53 

How corrected, 

Mural Circle, its object, 54 

Describe it, 54 

How the different parts contribute to 

theobject,. 54 

Mural Quadrant, 

Use of the Mural Circle for arcs of de- 
clination, 56 

Altitude and Azimuth Instrument de- 
fined,... , 56 

Page. Page. 

Its use, 56 

Describe it, 57 

Sextant described, 58 

How to measure the angular distance 

of the moon from the sun, 59 

How to take the altitude of a heavenly 

body - 59 

Use of the arti ficial horizon , 59 

In what consists the peculiar value of 

the Sextant? 60 

the sun's right ascension and decli- 
nation, to find his longitude and the 

obliquity of the ecliptic, 61 

Napier's Rule of circular parts, 62 

Given the sun's declination to find his 
rising and setting at any place whose 

latitude is known, 63 

Given the latitude of a place and the 
declination of a heavenly body, to 
determine its altitude and azimuth 
when on the six o'clock hour circle, 64 
The latitudes and longitudes of two 
celestial objects being given, to find 

their distance apart, 65 

reason for ascertaining it with great 

precision, 66 

How found from the centrifugal force, 66 
From measuring an arc of the meridian, 67 
From observations with the pendulum, 68 

From the motions of the moon, 68 

Density of the earth compared with 

water, 68 

How ascertained by Dr. Maskelyne, . . 69 
Why an important element, 69 



Figure of the sun, 70 

Angle subtended by a line of 400 miles, 70 

Distance from the earth, 70 

Illustrated by motion on a railway car, 70 
Apparent diameter of the sun new- 
found, 72 

How to find the linear diameter, 71 

ow much larger is the sun than the 

earth, 71 

53 Its density and mass compared with 

the earth's, 71 

Weight at the surface of the sun, 72 

52 H 

Velocity of falling bodies at the sun ... 72 

SOLAR SPOTS. Their number, 72 

55 Size, 72 

Description, 72 

What region of the sun do they oc- 
cupy, 73 

Proof that they are on the sun, 73 




How we learn the revolution of the sun 

on his axis, 73 

Time of the revolution, 73 

Apparent paths of the spots, 

Inclination of the solar axis, 74 

Sun's Nodes when does the sun pass 

them ? 75 

Cause of the solar spots, 76 

Faculae, 76 

ZODIACAL LIGHT. Where seen, 76 

Its form, 

Aspects at different seasons, 

Its motions, 

Its nature, 

Product of the angle described in any 
given time by the square of the dis- 
tance, 88 

74 Space described by the radius vector of 

the solar orbit in equal times, 88 

How to represent the sun's orbit by a 
diagram, 89 


77 H 



76 Universal Gravitation defined, 90 

76 Why is it called attraction, 90 

istory of its discovery, 90 

ow was the gravitation of the moon 

to the earth first inferred ? 91 

MOTION Laws of Gravitation. If a body re- 
volves about an immovable center 

77 H 

78 If 


Apparent motion of the sun , 

How both the sun and earth are said to 
move from west to east, 79 

Nature and position of the sun's orbit, 
how determined, 

Changes in declination how found, 79 

Ditto, in right ascension, 

Inferences from a table of the sun's de- 
clinations, 80 

Ditto, of right ascensions, 81 

Path of the sun, how proved to be a 
great circle, 81 

Obliquity of the ecliptic, how found, 81 

How it varies, 81 

Great dimensions of the earth's orbit, 81 

Earth's daily motion in miles, 

Ditto, hourly ditto, 

Diurnal motion at the equator per 

SEASONS. Causes of the change of sea- 

How each cause operates, 

Illustrated by a diagram, 

Change of seasons had the equator been 
perpendicular to the ecliptic, 84 

that the earth's orbit is not circular, 85 

Radius vector defined, 

Figure of the earth's orbit how ob- 

Relative distances of the earth from the 
sun, how found, 86 

Perihelion and Aphelion defined, 87 

Variations in the sun's apparent diame- 
ter,... . 87 

Angular velocities of the sun at the pe- 
rihelion and aphelion, 

Ratio of these velocities to the dis- 


How to calculate the relative distances 
of the earth from the sun's daily mo- 
tions, 88 


of force, and is constantly attracted 

to it, how will itmove? 92 

a body describes a curve around a 
center towards which it tends by any 
force, how is its angular velocity re- 
lated to the distance, 93 

n the same curve, the velocity at any 
point of the curve varies as what ? 93 

80 If equal areas be described about a cen- 
ter in equal times, to what must the 

force tend? 94 

How is the distance of any planet from 
the sun at any point in its orbit, to 
its distance from the superior focus ? 94 
Dase of two bodies gravitating to the 
same center where one descends in a 
straight line, and the other revolves 
in a curve, 95 

82 Velocity of a body at any point when 

falling directly to the sun, 97 

82 Relation between the distances and pe- 
riodic times, 99 

82 Kepler's three great laws, 99 


83 Idea of a projectile force, 100 

Mature of the impulse originally given 

to the earth, 100 

Two forces under which a body re- 
volves, 100 

85 Illustrated by the motion of a cannon 

ball, 101 

86 Why a planet returns to the sun, 102 

Illustration by a suspended ball, 103 


87 Precession of the Equinoxes defined, 104 
Why so called, 104 

87 Amount of Precession annually, 104 

Revolution of the equinoxes, 104 

Revolution of the pole of the equator 
around the pole of the ecliptic, 105 


Changes among the stars caused by 

precession, 105 

The present pole star not always such, 105 
What will be the pole star 13,000 

years hence ? 105 

Cause of the precession of the equi- 
noxes,... 105 

Explain how the cause operates, 

Proportionate effect of the sun and 
moon in producing precession, 



Tropical year defined, 107 

How much shorter than the sidereal 

year, 107 

Use of the precession of the equinoxes 

in chronology, 107 

NUTATION, defined, 108 

Explain its operation, 108 

Cause of Nutation, 108 

ABERRATION, defined, 108 



Direction of this motion, 110 

Illustrated by a diagram, . 

Amount of aberration, 

Effect on the places of the stars, 

MOTION OF THE APSIDES, the fact sta- 

Time of revolution of the line of Ap- 
sides, 110 

Present longitude of the perihelion,. . 110 

When was it nothing ? 110 


Mean Motion defined, Ill 

Illustrated by surveying a field, Ill 

Mean and true longitude distinguish- 



Equations defined, ........................ Ill 

Their object, ............................... Ill 

Mean and True anomaly defined, .... 112 

Equation of the Center, ................. 112 

Explain from the figure, ................ 112 




Distance of the moon from the earth, 

Her mean horizontal parallax, ......... 113 

Her diameter, .............................. 113 

Volume, density, and mass, ............ 113 

Shines by reflected light, ................ 113 

Appearance in the telescope, ........... 113 

Terminator defined, ...................... 113 

Its appearanpe, ............................ 113 

Proofs of Valleys, ......................... 114 

Form of these ............................. 114 

Best time for observing the lunar 

mountains and valleys, ............... 114 

Names of places on the moon double, 115 

Dusky regions how named, ............. 115 

Point out remarkable places on the 

map of the moon, ...................... 115 

Explain the method of estimating the 

Specify the heights of particular 

mountains, 117 

Volcanoes, proof of their existence,... . 117 

Has the moon an atmosphere ? 117 

Improbability of identifying artificial 

structures in the moon, 117 

PHASES OF THE MOON, their cause,.... 118 
Successive appearances of the moon 
from one new moon to another, .... 118 

Syzygies defined, 118 

Explain the phases of the moon from 

figure 46, 119 

of her revolutions about the earth, . 119 

Her apparent orbit a great circle, 120 

A sidereal month defined, 120 

A synodical do. 120 

Length of each, 120 

Why the synodical is longer, 120 

How eachis obtained, 120 

Inclination of the lunar orbit, 121 

Nodes defined, 121 

Why the moon sometimes runs high 

and sometimes low, 121 

Harvest moon defined, 122 

Ditto explained, 122 

Explain why the moon is nearer to us 
when on the meridian than when 

near the horizon, 122 

Time of the moon's revolution on its 

axis, 123 

How known, 123 

Librations explained, 123 

Diurnal Libration, 124 

Length of the Lunar days, 124 

Earth never seen on the opposite side 

of the moon, 124 

Appearances of the earth to a specta- 
tor on the moon, 124 

Why the earth would appear to re- 

main fixed, 125 

Ascending and descending nodes dis- 
tinguished, 125 

Whether the earth carries the moon 

around the sun, 126 

How much more is the moon attract- 
ed towards the sun than towards 

the earth, 126 

When does the sun act as a disturbing 

force upon the moon ? 126 

Why does not the moon abandon the 

earth at the conjunction ? 126 

The moon's orbit concave towards the 

sun, 127 

How the elliptical motion of the moon 
about the earth is to be conceived 

of, 127 

Illustrations, 127 


height of lunar mountains, 115 Specify their general cause, 127 


Unequal action of the sun upon the 
earth and moon, 

Oblique action of earth and sun, 

Gravity of the moon towards the 
earth at the syzygies, 

Gravity at the quadratures, 

Explain the disturbances in the 
moon's motions from figure 48, 

Figure of the moon's orbit, 

How its figure is ascertained, 

Moon's greatest and least apparent di- 

Her greatest and least distances from 
the earth, 

Perigee and Apogee defined, 

Eccentricities of the solar and lunar 
orbits compared, 

Moon's nodes, their change of place,. 

Rate of this change per annum, , 

Period of their revolution, 

Irregular curve described by the 

Cause of the retrograde motion of 

Explain from figure 50, 

Synodical revolution of the node de- 

Its period, 

The Saros explained, 

The Metonic Cycle, 

Golden Number, 

Revolution of the line of apsides, 

Its period, 

How the places of the perigee may be 

Moon's anomaly defined, 

Cause of the revolution of the apsides, 

Amount of the equation of the Center, 

Evection defined, 

Its cause explained, 

Variation defined, 

Its cause, 

Annual Equation explained, 

How these irregularities were first 

How many equations are applied to 
the moon's motions ? 

Method of proceeding in finding the 
moon's place, 

Successive degrees of accuracy at- 

Periodic and secular irregularities dis- 

Acceleration of the moon's mean mo- 
tion explained, 

Its consequences, 

Lunar inequalities of latitude and 



Eclipse of the moon, when it happens, 













Eclipse of the sun, when it happens, . 143 

When only can each occur, 143 

Why an eclipse does not occur at 

every new and full moon, 144 

Why eclipses happen at two opposite 

months, 144 

Circumstances which affect the length 

of the earth's shadow, 144 

Semi-angle of the cone of the earth's 

shadow, to what equal, 145 

Length of the earth's shadow, 145 

Its breadth where it eclipses the 

moon, 146 

Lunar ecliptic limit defined, 146 

Solar, ditto 146 

Amount of thelunar ecliptic limit,.... 146 

Appulse defined, 147 

Partial, total, central, eclipse, each 

defined, 147 

Penumbra defined,. 147 

Semi-angle of the moon's penumbra, 

to what equal, 148 

Semi-angle of a section of the penum- 
bra where the moon crosses it, 148 

Moon's horizontal parallax increased 

,why, 148 

Why the moon is visible in a total 

eclipse, 148 

alculation of eclipses, general mode 
of proceeding, 149 

To find the exact time of the begin- 
ning, end, duration, and magnitude 
of a lunar eclipse, by figures 53, 54, 150 

Elements of an eclipse defined, 151 

Digits defined, 153 

How the shadow of the moon travels 
over the earth in a solar eclipse,;... 153 

Why the calculation of a solar eclipse 

is more complicated than a lunar,. 154 

Velocity of the moon's shadow, 154 

Different ways in which the shadow 
traverses the earth, according as 
the conjunction is near the node or 
near the limit, 155 

When do the greatest eclipses hap- 
pen ? 155 

ase in which the moon's shadow 
nearly reaches the earth, 156 

low far may the shadow reach be- 
yond the center of the earth ? 157 

reatest diameter of the moon's sha- 
dow where it traverses the earth,. . 157 
reatest portion of the earth's surface 
ever covered by the moon's penum- 
bra, 157 

Vloon's apparent diameter compared 
with the sun's, 158 

Annular eclipse, its cause, 158 

Direction in which the eclipse passes 
on the sun's disk, 159 



Greatest duration of total darkness,. . . 15! 

Eclipses of the sun more frequent 
than of the moon, why ? 15! 

Lunar eclipses oftener visible, why ? 15! 

Radiation of light in a total eclipse of 
the sun, 16( 

Interesting phenomena of a total 
eclipse of the sun, 16( 

Phenomena of the eclipse of 1806, de- 
scribed, 16( 

When does the next total eclipse of 
the sun, visible in the United 
States, occur? 16 


Objects of the ancients in studying 

astronomy, 16 

Ditto of the moderns, 16 

LONGITUDE. How to find the differ- 
ence of longitude between two 
places, 161 

Method by the Chronometer explain- 
ed, 162 

How to set the chronometer to Green- 

wich time, 162 Inferior and superior planets distin- 

Accuracy of some chronometers, 162 


Longitude by eclipses explained, 

Lunar method of finding the longi- 

Circumstances which render this 

method somewhat difficult, 164 

Disadvantages of this method, 

Degree of accuracy attainable, 165 

TIDES. defined, 165 

High, Low, Spring, Neap, Flood, and 

Ebb Tide, severally defined, 165 

Similar tides on opposite sides of the 

earth, 165 

Interval between two successive high 

tides...... 165 

Average height for the whole globe, 166 

Extreme height, 166 

Cause of the tides, 166 

Explain by figure 56, 166 

Tide-wave defined, 167 

Comparative effects of the sun and 

moon in raising the tide, 167 

Why the moon raises a higher tide 

than the sun, 167 

Springtides accounted for, 168 

Neap tides, ditto 168 

Power of the sun or moon to raise 
the tide, in what ratio to its dis- 
tance, 168 

Influence of the declinations of the 

sun and moon on the tides, 169 



Cotidal Lines defined, 170 

Derivative and Primitive tides distin- 
guished, 170 

Velocity of the tide-wave, circum- 
stances which affect it, 171 

Explain by figure 59, 171 

Examples of very high tides, 172 

Unit of altitude defined, 172 

Unit of altitude for different places, 172 
Tides on the coast of N. America, 

whence derived, 173 

Why no tides in lakes and seas, 173 

Intricacy of the problem of the tides, 173 

Atmospheric tide, 173 


Etymology of the word planet, 174 

Planets known from antiquity, 174 

Ditto, recently discovered, 174 

Primary and Secondary Planets dis- 
tinguished, 174 

Whole number of each, 175 

Inclination of their orbits, 175 

guished, 175 

Objections to them, 162 Differences among the planets, 175 

- Distances from the sun in miles, 175 

Great dimensions of the planetary 

orbits, 176 

Flow long a railway car would re- 
quire to cross the orbit of Uranus, 176 

Law of the distances, 176 

Mean distances, how determined,.... 176 

Diameters in. miles, 177 

Flow ascertained, 177 

Per io die t imes, 178 

Which planets move most rapidly,... 178 
INFERIOR PLANETS. Their proximity 

to the sun, 178 

[llustrate by figure 60, 179 

When is a planet said to be in con- 

inferior and 


superior conjunctions 
distinguished, 179 

Synodical revolution of a planet de- 
fined, 179 

r hy its period exceeds that of the 
planet in its orbit, 179 

To ascertain the synodical period 

from the sidereal, 180 

lynodical periods of Mercury and 
Venus, 180 

Motions of an inferior planet de- 
scribed, 180 

xplain from figure 60, 180 

When is an inferior planet station- 



Explain from figures 57 and 58, 

Motion of the tide-wave not progres- Elongation of Mercury when sta- 

sive, 170 tionary, ". 181 

Tides of rivers, narrow bays, how [Ditto of Venus, 181 

produced, 170JPhases of Mercury and Venus, 182 



Distance of an inferior planet from Motions of the satellites, 

the sun, how found, 182 Diameter, 193 

Eccentricity of the orbit of Mercury, 182 Distances from the primary, 193 

Ditto of Venus, 182;Figure of their orbits, 194 

Most favorable time for determining Their inclination to the planet's equa- 

the sidereal revolution of a planet, 183j tor, 194 

When is an inferior planet brightest, 183 Their eclipses, how they differ from 
Times of their revolutions on their 

axes, 183 

Venus, her brightness, 183 

Her conjunctions with the sun every 
eight years, 184 

defined, 184 

Why a transit does not occur at 
every inferior conjunction, 184 

Why the transits of Mercury occur in 
May and November, and those of 
Venus in June and December, 185 

Intervals between successive transits, 
how found, 185 

Why transits of Venus sometimes oc- 
cur after an interval of eight years, 186 

Why transits are objects of so much 

interest, 186 SATURN, size 

Method of finding the sun's horizontal 

parallax from the transit of Venus, 187 
Why distant places are selected for 

observing it, 187 

Explain the principle from figure 61, 187 
Amount of the sun's horizontal par- 
allax, 188 

Indications of an atmosphere in Venus, 189 

Whether Venus has any satellite,... . 189 

Mountains of Venus, 189 


How the superior planets are distin- 
guished from the inferior, 189 

MARS, diameter, 190 

Mean distance from the sun, 190 

from those of the moon, 194 

Their phenomena explained from fig- 
ure 63, 195 

Shadow seen traversing the disk of 

the primary, 196 

Satellite itself seen on the disk, 196 

Remarkable relation between the 
mean motions of the three first 

satellites, 196 

Consequences of this, 196 

Use of the eclipses of Jupiter's satel- 
lites in finding the longitude, 197 

How it is adapted to this purpose,... 197 

Imperfections of this method, 197 

Why not practised at sea, 198 

Discovery of the progressive motion 
of light, how made, 198 


Number of satellites, 199 

Ring, double, 199 

Dimensions of the ring in several par- 
ticulars, 199 

Representation in the figure, (frontis- 
piece,) 199 

Proof that the ring is solid and opake, 199 
Proof that the axis of rotation is per- 
pendicular to the plane of the ring, 199 

Period of rotation, 200 

Compression of the poles, 200 

Peculiar figure of the planet, 200 

Parallelism of the ring in all parts of 

its revolution, 200 

Illustration by a small disk and a 

lamp, 200 

Different appearances of the ring, 200 

Explain diagram 64, 201 

Inclination of his orbit, 190 Proof that the ring shines by reflected 

Variation of brightness and magnitude, 
Explain the cause from figure 62,.... 


Phases of Mars, ~ 191 

Telescopic appearance, 191 

Revolution on his axis, 191 

Spheroidal figure, '. 

JUPITER great size, diameter, 191 

Spheroidal figure, 192 

Rapid diurnal revolution, 192 

Distance from the sun, 192 

Periodic time, 192 

Telescopic appearance, 192 

Why astronomers regard Jupiter and 

his moons with so much interest,... 192 

Belts, number, situation, cause, 192 

Satellites of Jupiter, number, situa- 

tion, 193 Irregular motions,. 

light, 202 

Revolution of the ring, 202 

How ascertained, 202 

Rings not concentric with the planet, 203 

Advantages of this arrangement, 203 

Appearance of the rings from the 

planet, 203 

Satellites, distance of the outermost 

from the planet, 204 

Description of the satellites, 204 

URANUS, distance and diameter, 204 

Period of revolution, inclination of its 

orbit, . 


His'tory of its discovery, 205 

Satellites, number, minuteness, 205 

Appearance of the sun from Uranus, 





NEW PLANETS, their names, 20f 

Position in the system, 206 

Discovery, 206 

Theoretic notions respecting their ori- 
gin, 206 

Reason of .their names, 207 

Their average distance, 207 

Periodic Times, 207 

Inclinations of their orbits, 207 

Eccentricity of do 20 

Small size, 208 

Atmospheres, 208 


Reasons for delaying the consideration 
of the planetary motions, 208 

Two methods of studying the celestial 
motions, 208 

Notions of absolute space, 209 

Appearance of the planets from the 
sun, 209 

Particular appearance of the orbit of 
Mercury, 21C 

Mutual relation of the orbit of the 
earth and Mercury considered, 210 

How the motions of the other plan, 
ets differ from from those of Mer- 
cury, 210 

Why is the ecliptic taken as the stan- 
dard of reference, 210 

Three particulars necessary in order 
to represent the actual positions of 
the planetary orbits, 211 

Why diagrams represent the orbits er- 
roneously, as figure 65, 211 

Inadequate representations of the so- 
lar system, 213 

How the planets would be truly repre- 
sented, 213 

Two reasons why the apparent mo- 
tions are unlike the real, 213 

Explain figure 66, 214 

Motions of Venus compared with those 
of Mercury, 215 

Apparent motions of the superior plan- 
ets, how far they are like and how 
unlike the inferior, 215 

Explain figure 67, 215 


Ptolemy's views of the figure of the 
planetary orbits, 217 

Kepler's investigation of the motions 
of Mars, 218 


History of the discovery O f Kepler's 
Laws, 218 

Third Law, how modified by the quan- 
tity of matter, 219 

ELEMENTS, their number, 220 

Enumeration of them, 220 

Why we cannot find them as we do 
those of the moon and sun, 220 

First steps in the process, 221 

To find the heliocentric longitude and 
latitude of a planet, figure 68, 221 

To find the position of the nodes, and 
the inclination, figure 69, 222 

To find the periodic time, 223 

Difficulty of finding when a planet is 
at itsnode, 223 

Advantage of observations taken when 
a planet is in opposition, 223 

Periodic time, how ascertained most 
accurately, 224 

To find the major axis of the planet- 
ary orbits, 224 

onstancy of the major axis, 225 

To find the place of the perihelion, 
figure 71, 226 

To find the place of the planet in its 
orbit at a particular epoch, 227 

To find the eccentricity, - . 227 

AND PLANETS. How we learn the 
quantity of matter in a body, 22b 

Method by means of the distances and 
periodic times of their satellites, 228 

Vlass of the sun compared with that of 
the earth, 229 

Same result how deduced from the 
centrifugal force, 229 

Mass of the planets that have no sat- 
ellites how found, 229 

flow the quantity of matter in bodies 

varies 230 

their densities vary, 230 

Inferences from the table of densities 
and specific gravities of the planets, 230 

Perturbations produced by the planets, 231 

Probability of derangement in the 
planetary motions, 231 

Actual changes, 231 

Jesuit of the investigations of La Place 
and La Grange, 232 

mportant relation between great 
masses and small eccentricities,... . 233 

Chapter XIII. COMETS. 

Three parts of a comet 234 

Description of each part, 234 

Number of Comets, 234 

Six particularly remarkable 235 

Differences in magnitude and bright- 

ness, 236 




Variations in the same comet at dif- 
ferent returns, 236 

Periods of comets, 237 

Distances of their aphelia, 237 

Proof that they shine by reflected 

light, 237 

Changes in the tail at different dis- 
tances from the sun, 237 

Direction from the sun, 237 

Quantity of Matter, 238 

Effect when they pass very near the 

planets, 238 

Proof that they consist of matter, 238 
How a comet's orbit may be entirely 

changed, 239 

How exemplified in the comet of 1770, 239 

Nature of their Orbits, 240 

Five Elements of a Comet, 240 

Investigation of these elements, why 

so difficult, 241 

Can the length of the major axis be 

calculated? 242 

How determined, 242 

Elements of a comet, how calculated 243 
How a comet is known to be the same 

as one that has appeared before,... . 243 

Exemplified in Halley's comet, 243 

Return in 1835, 244 

Encke's comet, appearance in 1839, 244 

Proofs of a Resisting Medium, 244 

Its consequences, 244 

Physical nature of comets, 245 

How their tails are supposed to be 

formed, 245 

Difficulty of accounting for the direc- 
tion of the tail, 245 

Supposition of Delambre, 246 

Possibility of a comet's striking the 

earth, 246 

Instances of comets coming near the 

earth, 247 

Consequences of a collision, 247 



Fixed stars, why so called, 248 

Magnitudes, how many visible to the 

naked eye, 248 

Whole number of magnitudes, 248 

Antiquity of the constellations, 248 

Whether the names are founded on 

resemblance, 249 

Names of the individual stars of a con- 
stellation, 249 

Catalogues of the stars, 249 

Numbers in different catalogues, 249 CLUSTERS 

Utility of learning the constellations, 250 


CONSTELLATIONS. Aries, how recog- 
nized, 251 

Taurus do. largest star in Taurus, ... 251 
Gemini, magnitude of Castor, of Pol- 
lux, 251 

Cancer, size of its stars, Prssepe, . . . . 251 
Leo, size, magnitude of Regulus, sickle, 

Denebola, 251 

Virgo, direction, Spica, Vindemiatrix, 252 

Libra, how distinguished, 252 

Scorpio, his head how formed, An- 

tares, tail, 252 

Sagittarius, direction from Scorpio, 

how recognized, 252 

! apricornus, direction from Sagitta- 
rius, t\vo stars, 

Aquarius, its shoulders, , 252 

Pisces, situation, 252 

Piscis Australis, Fomalhaut, 252 

Andromeda, how characterized, Mi- 

rach, Almaak, ; 253 

Perseus, Algol, Algenib, 253 

Auriga, situation, Capella, its mag- 
nitude, 253 

Lynx 253 

Leo Minor, situation from Leo, 253 

oma Berenices, direction from Leo, 

CorCaroli 253 

Bootes, Arcturus, size and color, 253 

orona Borealis, where from Bootes, 

figure, 254 

Hercules, number of stars, great extent, 254 

Ophiuchus, where from Hercules, 254 

Aquila, three stars, Altair, Antinous, 254 

Delphinus, four stars, tail, 254 

Pegasus, four stars in a square, their 

names, 254 

Ursa Minor, Pole-star, Dipper, 254 

Ursa Major, how recognized, Point- 
ers, Alioth, Mizar, 255 

Draco, position with respect to the 

Great and Little Bear, 255 

Cepheus, where from the Dragon, size 

of its stars, 255 

assiopeia's chair in the Milky Way, 255 
Cygnus, where from Cassiopeia, fig- 
ure, 255 

Lyra, largest star, 255 

Cetus, its extent, Menkar, Mira, 256 

Orion, size and beauty, parts, 256 

Canis Major, where from Orion, Sirius, 256 

Canis Minor, Procyon, 256 

Hydra situation Cor Hydrae, 256 

"orvus, how represented, 256 


. Examples, 257 

Number of stars in the Pleiades, 257 




Stars of Coma Berenices and Pree- 
sepe 257 

260 Di 

NEBULA. Defined, 258 

Examples, 258 

Number in Herschel's Catalogue, 258 

Herschel's Views of their nature, 

Figures of nebulae, 259 

Nebula in the Sword of Orion, 259 

Nebulous Stars, defined, figures, 

Annular Nebula, appearance, exam- 


Galaxy or Milky Way, HerschePs 

views of it, 

VARIABLE STARS. Defined, 260 

Examples in o Ceti and Algol, 261 

TEMPORARY STARS. -Defined, 261 

Examples, why Hipparchus number- 
ed the stars, 261 

Stars seen by the ancients, now miss- 
ing, .' 261 

DOUBLE STARS. Defined, examples,. 262 
Distance between the double stars in 


Colors of some double stars 262 

Examples, 263 

Number, 263 


BINARY STARS, how distinguished 
from common double stars, 264 

Examples of revolving stars, 

Inferences from the tabular view, 

Particulars of y Virginis, 

Proof that the law of gravitation ex- 
tends to the stars, 

Whether these stars are of a planeta- 
ry or a cometary nature, 267 

PROPER MOTIONS. Result on com- 
paring the places of certain stars 
with those they had in the time of 
Ptolemy, 267 

Conclusion respecting the apparent 
motions of certain stars, 267 

How the fact of the sun's motion 
might be proved, 267 

Example of stars having a proper mo- 
tion, 267 

What class of stars have the greatest 
proper motion? 


What we can determine respecting 
the distance of the nearest star,.... 268 

Base line for measuring this distance, 269 

Have the stars any parallax ? 269 

259 Taking the parallax at 1", find the 

distance, 269 

Amount of this distance, 269 

259 Probable greater distance of the 

smaller stars, 270 

isputes respecting the parallax of 

the stars, 270 

260 To find the parallax by means of the 

double stars, 270 

Minuteness of the angles estimated, . 270 
How the magnitude of the stars is 

affected by the telescope, 271 


compared with the earth, 271 

Dr. Wollaston's observation on their 

comparative light, 271 

Proofs that they are suns, 272 

262 Arguments for a Plurality of Worlds, 272 


System of the world defined, 273 

Compared to a machine, 273 

FIXED Astronomical knowledge of the an- 
cients, 373 

Things known to Pythagoras, 273 

His views of the system of the world, 273 

yrstalline Spheres of Eudoxus, 274 

265 Knowledge possessed by Hipparchus, 275 

265 Ptolemaic System, 276 

265 Deferents and epicycles defined, 276 

Explained by figure 74, 276 

266 How far this system would explain 

the phenomena, 277 

Its absurdities, 277 

Objections to the Ptolemaic System,. 278 

Tychonic System, 278 

Its advantages, 278 

Its absurdities, 278 

Copernican System, 279 

Proofs that the earth revolves, 279 

Ditto, that the planets revolve about 

the sun, 279 

Higher orders of relations among the 

stars, 280 

Proofs of such orders, 280 

268 Structure of the material universe,.. 281 

DCr* Diagrams for public recitations. 

As many of the figures of this work are too complicated to be 
drawn on the black-board at each recitation, we have found it 
very convenient to provide a set of permanent cards of paste- 
board, on which the diagrams are inscribed on so large a scale, as 
to be distinctly visible in all parts of the lecture room. The let- 
ters may be either made with a pen, or better procured of the 
printer, and pasted on. 

The cards are made by the bookbinder, and consist of a thick 
paper board about 18 by 14 inches, on each side of which a white 
sheet is pasted, with a neat finish around the edges. A loop at- 
tached to the top is convenient for hanging the card on a nail. 

Cards of this description, containing diagrams for the whole 
course of mathematical and philosophical recitations, have been 
provided in Yale College, and are found a valuable part of our ap- 
paratus of instruction. 

U~p Several valuable articles, not contained in preceding edi- 
tions, will be found in the Addenda. Article IV., on the Nume- 
rical Relations existing between the Members of the Solar System, 
with problems, is particularly recommended to students in Astron- 
omy. Notices of recent discoveries will be found in Article V. ; 
and at the end of the volume is inserted a Syllabus of the Lectures 
on Astronomy, delivered to the students in Yale College after they 
have perused this treatise. 



1. ASTRONOMY is that science which treats of the heavenly bodies. 

More particularly, its object is to teach what is known respect- 
ing the Sun, Moon, Planets, Comets, and Fixed Stars ; and also to 
explain the methods by which this knowledge is acquired. Astron- 
omy is sometimes divided into Descriptive, Physical, and Practi- 
cal. Descriptive Astronomy respects facts ; Physical Astronomy. 
causes; Practical Astronomy, the means of investigating the facts, 
whether by instruments, or by calculation. It is the province of 
Descriptive Astronomy to observe, classify, and record, all the 
phenomena of the heavenly bodies, whether pertaining to those 
bodies individually, or resulting from their motions and mutual 
relations. It is the part of Physical Astronomy to explain the 
causes of these phenomena, by investigating and applying the 
general laws on which they depend ; especially by tracing out all 
the consequences of the law of universal gravitation. Practical 
Astronomy lends its aid to both the other departments. 

2. Astronomy is the most ancient of all the sciences. At a 
period of very high antiquity, it was cultivated in Egypt, in Chal- 
dea, in China, and in India. Such knowledge of the heavenly 
bodies as could be acquired by close and long continued observa- 
tion, without the aid of instruments, was diligently amassed ; and 
tables of the celestial motions were constructed, which could be 
used in predicting eclipses, and other astronomical phenomena. 

About 500 years before the Christian era, Pythagoras, of 
Greece, taught astronomy at the celebrated school at Crotona, and 
exhibited more correct views of the nature of the celestial mo- 
tions, than were entertained by any other astronomer of the an- 
cient world. His views, however, were not generally adopted, 



but lay neglected for nearly 2000 years, when they were revived 
and established by Copernicus and Galileo. The most celebrated 
astronomical school of antiquity was at Alexandria, in Egypt, 
which was established and sustained by the Ptolemies, (Egyptian 
princes,) about 300 years before the Christian era. The employ- 
ment of instruments for measuring angles, and the introduction of 
trigonometrical calculations to aid the naked powers of observa- 
tion, gave to the Alexandrian astronomers great advantages over 
all their predecessors. The most able astronomer of the Alexan- 
drian school was Hipparchus, who was distinguished above all the 
ancients for the accuracy of his astronomical measurements and 
determinations. The knowledge of astronomy possessed by the 
Alexandrian school, and recorded in the Almagest* or great work 
of Ptolemy, constituted the chief of what was known of our 
science during the middle ages, until the fifteenth and sixteenth 
centuries, when the labors of Copernicus of Prussia, Tycho Brake 
of Denmark, Kepler of Germany, and Galileo of Italy, laid the 
solid foundations of modern astronomy. Copernicus expounded 
the true theory of the celestial motions ; Tycho Brahe carried 
the use of instruments and the art of astronomical observation to 
a far higher degree of accuracy than had ever been done before ; 
Kepler discovered the great laws of the planetary motions ; and 
Galileo, having first enjoyed the aid of the telescope, made innu- 
merable discoveries in the solar system. Near the beginning of 
the eighteenth century, Sir Isaac Newton discovered, in the law 
of universal gravitation, the great principle that governs the ce- 
lestial motions ; and recently, La Place has more fully completed 
what Newton began, having followed out all the consequences of 
the law of universal gravitation, in his great work, the Mecan- 
ique Celeste. 


3. Among the ancients, astronomy was studied chiefly as sub- 
sidiary to astrology. ASTROLOGY was the art of divining future 
events by the stars. It was of two kinds, natural and judicial. 
Natural Astrology, aimed at predicting remarkable occurrences in 
the natural world, as earthquakes, volcanoes, tempests, and pesti- 
lential diseases. Judicial Astrology, aimed at foretelling the fates 
of individuals, or of empires. 


4. Astronomers of every age, have been distinguished for their 
persevering industry, and their great love of accuracy. They 
have uniformly aspired to an exactness in their inquiries, far be- 
yond what is aimed at in most geographical investigations, satis- 
fied with nothing short of numerical accuracy, wherever this is 
attainable ; and years of toilsome observation, or laborious calcu- 
lation, have been spent with the hope of attaining a few seconds 
nearer to the truth. Moreover, a severe but delightful labor is 
imposed on all who would arrive at a clear and satisfactory knowl- 
edge of the subject of astronomy. Diagrams, artificial globes, 
orreries, and familiar comparisons and illustrations, proposed by 
the author or the instructor, may afford essential aid to the learner, 
but nothing can convey to him a perfect comprehension of the 
celestial motions, without much diligent study and reflection. 

5. In expounding the doctrines of astronomy, we do not, as in 
geometry, claim that every thing shall be proved as soon as as- 
serted. We may first put the learner in possession of the leading 
facts of the science, and afterwards explain to him the methods 
by which those facts were discovered, and by which they may 
be verified ; we may assume the principles of the true system of 
the world, and employ those principles in the explanation of many 
subordinate phenomena, while we reserve the discussion of the 
merits of the system itself, until the learner is extensively ac- 
quainted with astronomical facts, and therefore better able to ap- 
preciate the evidence by which the system is established. 

6. The Copernican System is that which is held to be the true 
system of the world. It maintains (1,) That the apparent diur- 
nal revolution of the heavenly bodies, from east to west, is owing 
to the real revolution of the earth on its own axis from west to 
east, in the same time ; and (2,) That the sun is the center around 
which the parth and planets all revolve from west to east, con- 
trary to the opinion that the earth is the center of motion of the 
sun and planets. 

7. We shall treat, first, of the Earth in its astronomical rela- 
tions ; secondly, of the Solar System ; and, thirdly, of the Fixed 




8. The figure of the earth is nearly globular. This fact is 
known, first, by the circular form of its shadow cast upon the 
moon in a lunar eclipse ; secondly, from analogy, each of the 
other planets being seen to be spherical ; thirdly, by our seeing 
the tops of distant objects while the other parts are invisible, as 
the topmast of a ship, while either leaving or approaching the 
shore, or the lantern of a light-house, which, when first descried 
at a distance at sea, appears to glimmer upon the very surface of 
the water ; fourthly, by the depression or dip of the horizon when 
the spectator is on an eminence ; and, finally, by actual observa- 
tions and measurements, made for the express purpose of ascer- 
taining the figure of the earth, by means of which astronomers are 
enabled to compute the distances from 

the center of the earth of various places 
on its surface, which distances are found 
to be nearly equal. 

9. The Dip of the Horizon, is the ap- 
parent angular depression of the hori- 
zon, to a spectator elevated above the 
general level of the earth. The eye 
thus situated, takes in more than a ce- 
lestial hemisphere, the excess being the 
measure of the dip. 

Thus, in Fig. 1, let AO represent the 


height of a mountain, ZO the direction of the plumb line, HOR a 
line touching the earth at the point O,' and at right angles to the 
plumb line, C the center of the earth, DAE the portion of the 
earth's surface seen from O; OD, OE, lines drawn from the 
place of the spectator to the most distant parts of the horizon, 
and CD a radius of the earth. The dip of the horizon is the an- 
gle HOD or ROE. Now the angle made between the direction 
of the plumb line and that of the extreme line of the horizon or 
the surface of the sea, namely, the angle ZOD, can be easily 
measured ; and subtracting the right angle ZOH from ZOD, the 
remainder is the dip of the horizon, from which the length of the 
line OD may be calculated, (see Art. 10,) the height of the spec- 
tator, that is, the line OA, being known. This length, to whatever 
point of the horizon the line is drawn, is always found to be the 
same ; and hence it is inferred, that the boundary which limits 
the view on all sides, is a circle. Moreover, at whatever elevation 
the dip of the horizon is taken, in any part of the earth, the 
space seen by the spectator is always circular. Hence the sur- 
face of the earth is spherical. 

10. The earth being a sphere, the dip of the horizon HOD= 
OCD. Therefore, to find the dip of the horizon corresponding 
to any given height AO* (the diameter of the earth being known,) 
we have in the triangle OCD, the right angle at D, and the two 
sides CD, CO, to find the angle OCD. Therefore, 

CO : rad. : : CD : cos. OCD. Learning the dip corresponding 
to different altitudes, by giving to the line AO different values, 
we may arrange the results in a table. 

* The learner will remark that the line AO, as drawn in the figure, is much larger 
m proportion to CA than is actually the case, and that the angle HOD is much too 
great for the reality. Such disproportions are very frequent in astronomical diagrams, 
especially when some of the parts are exceedingly small compared with others ; and 
hence the diagrams employed in astronomy are not to be regarded as true pictures of 
the magnitudes concerned, but merely as representing their abstract geometrical re. 


Table showing the Dip of the Horizon at different elevations, from 
I foot to WO feet* 


/ // 


/ // 


/ // 









































































Such a table is of use in estimating the altitude of a body 
above the horizon, when the instrument (as usually happens) is 
more or less elevated above the general level of the earth. For 
if it is a star whose altitude above the horizon is required, the 
instrument being situated at O, (Fig. 1,) the inquiry is how far 
the star is elevated above the level HOR, but the angle taken is 
that above the visible horizon OD. The dip, therefore, or the 
angle HOD, corresponding to the height of the point O, must be 
subtracted, to obtain the true altitude. On the Peak of Tene- 
rifte, a mountain 13,000 feet high, Humboldt observed the surface 
of the sea to be depressed on all sides nearly 2 degrees. The 
sun arose to him 12 minutes sooner than to an inhabitant of the 
plain ; and from the plain, the top of the mountain appeared en- 
lightened 12 minutes before the rising or after the setting of 
the sun. 

11. The foregoing considerations show that the form of the 
earth is spherical ; but more exact determinations prove, that the 
earth, though nearly globular, is not exactly so : its diameter from 
the north to the south pole is about 26 miles less than through 
the equator, giving to the earth the form of an oblate spheroid,! 

* This table includes the allowance for refraction. 

t An oblate spheroid is the solid described by the revolution of an ellipse about its 
shorter axis. 


or a flattened sphere resembling an orange. We shall reserve the 
explanations of the methods by which this fact is established, 
until the learner is better prepared than at present to understand 

12. The mean or average diameter of the earth, is 7912^4 miles, 
a measure which the learner should fix in his memory as a stand- 
ard of comparison in astronomy, and of which he should endeavor 
to form the most adequate conception in his power. The circum- 
ference of the earth is about 25,000 miles (24857.5).* Although 
the surface of the earth is uneven, sometimes rising in high moun- 
tains, and sometimes descending in deep valleys, yet these eleva- 
tions and depressions are so small in comparison with the immense 
volume of the globe, as hardly to occasion any sensible deviation 
from a surface uniformly curvilinear. The irregularities of the 
earth's surface in this view, are no greater than the rough points 
on the rind of an orange, which do not perceptibly interrupt its 
continuity ; for the highest mountain on the globe is only about 
five miles above the general level ; and the deepest mine hitherto 

opened is only about half a mile.f Now i^r^r^;' or about 

one sixteen hundredth part of the whole diameter, an inequality 
which, in an artificial globe of eighteen inches diameter, amounts 
to only the eighty-eighth part of an inch. 

13. The diameter of the earth, con- 
sidered as a perfect sphere, may be de- 
termined by means of observations on 
a mountain of known elevation, seen 
in the horizon from the sea. Let BD 
(Fig. 2,) be a mountain of known 
height a, whose top is seen in the hori- 
zon by a spectator at A, b miles from it. 
Let x denote the radius of the earth. 
Then & + b 2 = (x+a)* = & + 2ax + a?. 

* It will generally be sufficient to treasure up in the memory the round number, 
but sometimes, in astronomical calculations, the more exact number may be required, 
and it is therefore inserted. 

t Sir John HerscheL 


7,2 2 

Hence, 2ax=tf*cP, and x= - . For example, suppose the 

height of the mountain is just one mile ; then it will be found, 
by observation, to be visible on the horizon at the distance of 

89 miles=6. Hence, z_-=:=3960= radius 

Zo, & 2 

of the earth, and 7920=the earth's diameter. 

14. Another method, and the most ancient, is to ascertain the 
distance on the surface of the earth, corresponding to a degree of 
latitude. Let us select two convenient places, one lying directly 
north of the other, whose difference of latitude is known. Sup- 
pose this difference to be 1 30', and the distance between the 
two places, as measured by a chain, to be 104 miles. Then, 
since there are 360 degrees of latitude in the entire circumference, 

1 30' : 104 : : 360 : 24960. And =7944. 

The foregoing approximations are sufficient to show that the 
earth is about 8,000 miles in diameter. 

15. The greatest difficulty in the way of acquiring correct 
views in astronomy, arises from the erroneous notions that pre- 
occupy the mind. To divest himself of these, the learner should 
conceive of the earth as a huge globe occupying a small portion 

Fig. 3. 


of space, and encircled on all sides with the starry sphere. He 
should free his mind from its habitual proneness to consider one 
part of space as naturally up and another down, and view him- 
self as subject to a force which binds him to the earth as truly as 
though he were fastened to it by some invisible cords or wires, as 
the needle attaches itself to all sides of a spherical loadstone. He 
should dwell on this point until it appears to him as truly up in 
the direction of BB , CC , DD , (Fig. 3,) when he is at B, C, and 
D, respectively, as in the direction of AA when he is at A. 


16. The definitions of the different lines, points, and circles, 
which are used in astronomy, and the propositions founded upon 
them, compose the Doctrine of the Sphere.* 

17. A section of a sphere by a plane cutting it in any manner, 
is a circle. Great circles are those which pass through the center 
of the sphere, and divide it into two equal hemispheres : Small 
circles, are such as do not pass through the center, but divide the 
sphere into two unequal parts. Every circle, whether great or 
small, is divided into 360 equal parts called degrees. A degree, 
therefore, is not any fixed or definite quantity, but only a certain 
aliquot part of any circle. 

18. The Axis of a circle, is a straight line passing through its 
center at right angles to its plane. 

19. The Pole of a great circle, is the point on the sphere where 
its axis cuts through the sphere. Every great circle has two 
poles, each of which is every where 90 from the great circle. 
For, the measure of an angle at the center of a sphere, is the 
arc of a great circle intercepted between the two lines that con- 
tain the angle ; and, since the angle made by the axis and any 
radius of the circle is a right angle, consequently its measure on 
the sphere, namely, the distance fropa the pole to the circumfer- 

* It is presumed that many of those who read this work, will have studied Spherical 
Geometry ; but it is so important to the student of astronomy to have a clear idea of 
the circles of the sphere, that it is thought best to introduce them here. 


ence of the circle, must be 90. If two great circles cut each 
other at right angles, the poles of each circle lie in the circum- 
ference of the other circle. For each circle passes through the 
axis of the other. 

20. All great circles of the sphere cut each other in two points 
diametrically opposite, and consequently, their points of section 
are 180 apart. For the line of common section, is a diameter 
in both circles, and therefore bisects both. 

21. A great circle which passes through the pole of another 
great circle, cuts the latter at right angles. For, since it passes 
through the pole and the center of the circle, it must pass through 
the axis ; which being at right angles to the plane of the circle, 
every plane which passes through it is at right angles to the same 

The great circle which passes through the pole of another great 
circle and is at right angles to it, is called a secondary to that circle. 

22. The angle made by two great circles on the surface of the 
sphere, is measured by the arc of another great circle, of which 
the angular point is the pole, being the arc of that great circle 
intercepted between those two circles. For this arc is the meas- 
ure of the angle formed at the center of the sphere by two radii, 
drawn at right angles to the line of common section of the two 
circles, one in one plane and the other in the other, which angle 
is therefore that of the inclination of those planes. 

23. In order to fix the position of any plane, either on the sur- 
face of the earth or in the heavens, both the earth and the heav- 
ens are conceived to be divided into separate portions by circles, 
which are imagined to cut through them in various ways. The 
earth thus intersected is called the terrestrial, and the heavens the 
celestial sphere. The learner will remark, that these circles have 
no existence in nature, but are mere landmarks, artificially con- 
trived for convenience of reference. On account of the immense 
distance of the heavenly bodies, they appear to us, wherever we 
are placed, to be fixed in the same concave surface, or celestial 


vault. The great circles of the globe, extended every way to 
meet the concave surface of the heavens, become circles of the 
celestial sphere. 


24. The Horizon is the great circle which divides the earth 
into upper and lower hemispheres, and separates the visible heav- 
ens from the invisible. This is the rational horizon. The sen- 
sible horizon, is a circle touching the earth at the place of the 
spectator, and is bounded by the line in which the earth and skies 
seem to meet. The sensible horizon is parallel to the rational, 
but is distant from it by the semi-diameter of the earth, or nearly 
4,000 miles. Still, so vast is the distance of the starry sphere, 
that both these planes appear to cut that sphere in the same line ; 
so that we see the same hemisphere of stars that we should see if 
the upper half of the earth were removed, and we stood on the 
rational horizon. 

25. The poles of the horizon are the zenith and nadir. The 
Zenith is the point directly over our head, and the Nadir that di- 
rectly under our feet. The plumb line is in the axis of the hori- 
zon, and consequently directed towards its poles. 

Every place on the surface of the earth has its own horizon ; 
and the traveller has a new horizon at every step, always extend- 
ing 90 degrees from his zenith in all directions. 

26. Vertical circles are those which pass through the poles of 
the horizon, perpendicular to it. 

The Meridian is that vertical circle which passes through the 
north and south points. 

The Prime Vertical, is that vertical circle which passes through 
the east and west points. 

27. As in geometry, we determine the position of any point by 
means of rectangular coordinates, or perpendiculars drawn from 
the point to planes at right angles to each other, so in astron- 
omy we ascertain the place of a body, as a fixed star, by taking 
its angular distance from two great circles, one of which is per- 
pendicular to the other. Thus the horizon and the meridian, or the 


horizon and the prime vertical, are coordinate circles used for such 

The Altitude of a body, is its elevation above the horizon meas- 
ured on a vertical circle. 

The Azimuth of a body, is its distance measured on the hori- 
zon from the meridian to a vertical circle passing through the body. 

The Amplitude of a body, is its distance on the horizon, from 
the prime vertical, to a vertical circle passing through the body. 

Azimuth is reckoned 90 from either the north or south point ; 
and amplitude 90 from either the east or west point. Azimuth 
and amplitude are mutually complements of each other. When a 
point is on the horizon, it is only necessary to count the number 
of degrees of the horizon between that point and the meridian, 
in order to find its azimuth ; but if the point is above the horizon, 
then its azimuth is estimated by passing a vertical circle through 
it, and reckoning the azimuth from the point where this circle cuts 
the horizon. 

The Zenith Distance of a body is measured on a vertical cir- 
cle, passing through that body. It is the complement of the alti- 

-28. The Axis of the Earth is the diameter, on which the earth 
is conceived to turn in its diurnal revolution. The same line con- 
tinued until it meets the starry concave, constitutes the axis of the 
celestial sphere. 

The Poles of the Earth are the extremities of the earth's axis: 
the Poles of the Heavens, the extremities of the celestial axis. 

29. The Equator is a great circle cutting the axis of the earth 
at right angles. Hence the axis of the earth is the axis of the 
equator, and its poles are the poles of the equator. The intersec- 
tion of the plane of the equator with the surface of the earth, 
constitutes the terrestrial, and with the concave sphere of the 
heavens, the celestial equator. The latter, by way of distinction, 
is sometimes denominated the equinoctial. 

30. The secondaries to the equator, that is, the great circles 
passing through the poles of the equator, are called Meridians 


because that secondary which passes through the zenith of any 
place is the meridian of that place, and is at right angles both to 
the equator and the horizon, passing as it does through the poles 
of both. (Art. 21.) These secondaries are also called Hour Circles, 
because the arcs of the equator intercepted between them are used 
as measures of time. 

31. The Latitude of a place on the earth, is its distance from 
the equator north or south. The Polar Distance, or angular dis- 
tance from the nearest pole, is the complement of the latitude. 

32. The Longitude of a place is its distance from some stand- 
ard meridian, either east or west, measured on the equator. The 
meridian usually taken as the standard, is that of the Observatory 
of Greenwich, near London. If a place is directly on the equator, 
we have only to inquire how many degrees of the equator there 
are between that place and the point where the meridian of Green- 
wich cuts the equator. If the place is north or south of the equa- 
tor, then its longitude is the arc of the equator intercepted between 
the meridian which passes through the place, and the meridian ot 

33. The Ecliptic is a great circle in which the earth perforate 
its annual revolution around the sun. It passes through the center 
of the earth and the center of the sun. It is found by observa- 
tion that the earth does not lie with its axis at right angles to the 
plane of the ecliptic, but that it is turned about 23 degrees out of 
a perpendicular direction, making an angle with the plane itself of 
66. The equator, therefore, must be turned the same distance 
out of a coincidence with the ecliptic, the two circles making 
an angle with each other of 23J, (23 27' 40".) It is particu- 
larly important for the learner to form correct ideas of the eclip- 
tic, and of its relations to the equator, since to these two circles a 
great number of astronomical measurements and phenomena are 

34. The Equinoctial Points, or Equinoxes* are the intersec- 

* The term Equinoxes strictly denotes the times when the sun arrives at the equi. 
noctial points, but it is also frequently used to denote those points themselves. 


tions of the ecliptic and equator. The time when the sun crosses 
the equator in returning northward is called the vernal, and in 
going southward, the autumnal equinox. The vernal equinox 
occurs about the 21st of March, and the autumnal the 22d of 

35. The Solstitial Points are the two points of the ecliptic 
most distant from the equator. The times when the sun comes 
to them are called solstices. The summer solstice occurs about 
the 22d of June, and the winter solstice about the 22d of De- 

The ecliptic is divided into twelve equal parts of 30 each, 
called signs, which, beginning at the vernal equinox, succeed each 
other in the following order : 

Northern. Southern. 

1. Aries T 7. Libra ^ 

2. Taurus 8 8. Scorpio tn, 

3. Gemini n 9. Sagittarius / 

4. Cancer S 10. Capricornus V3 

5. Leo 1 11. Aquarius ~ 

6. Virgo nj; 12, Pisces x 

The mode of reckoning on the ecliptic, is by signs, degrees, 
minutes, and seconds. The sign is denoted either by its name 
or its number. Thus 100 may be expressed either as the 10th 
degree of Cancer, or as 3 s 10. 

36. Of the various meridians, two are distinguished by the 
name of Colures. The Equinoctial Colure, is the meridian which 
passes through the equinoctial points. The Solstitial Colure, is 
the meridian which passes through the solstitial points. As the 
solstitial points are 90 from the equinoctial points, so the sol- 
stitial colure is 90 from the equinoctial colure. It is also at right 
angles, or a secondary to both the ecliptic and equator. For like 
every other meridian, it is of course perpendicular to the equator, 
passing through its poles. Moreover, the equinox, being a point 
both in the equator and in the ecliptic, is 90 from the solstice, 
from the pole of the equator, and from the pole of the ecliptic. 



Hence the solstitial colure, which passes through the solstice and 
the pole of the equator, passes also through the pole of the ecliptic, 
being the great circle of which the equinox itself is the pole. 
Consequently, the solstitial colure is a secondary to both the equa- 
tor and the ecliptic. (See Arts. 19, 20, 21.) 

37. The position of a celestial body is referred to the equator 
by its right ascension and declination. (See Art. 27.) Right 
Ascension, is the angular distance from the vernal equinox, meas- 
ured on the equator. If a star is situated on the equator, then its 
right ascension is the number of degrees of the equator between 
the star and the vernal equinox. But if the star is north or south 
of the equator, then its right ascension is the arc of the equator 
intercepted between the vernal equinox and that secondary to the 
equator which passes through the star. Declination is the dis- 
tance of a body from the equator, measured on a secondary to the 
latter. Therefore, right ascension and declination correspond to 
terrestrial longitude and latitude, right ascension being reckoned 
from the equinoctial colure, in the same manner as longitude is 
reckoned from the meridian of Greenwich. On the other hand, 
celestial longitude and latitude are referred, not to the equator, 
but to the ecliptic. Celestial Longitude, is the distance of a body 
from the vernal equinox reckoned on the ecliptic. Celestial Lati- 
tude, is distance from the ecliptic measured on a secondary to the 
latter. Or, more briefly, Longitude is distance on the ecliptic ; 
Latitude, distance from the ecliptic. The North Polar Distance 
of a star, is the complement of its declination. 

38. Parallels of Latitude are small 
circles parallel to the equator. They 
constantly diminish in size as we go 
from the equator to the pole, the ra- 
dius being always equal to the cosine 
of the latitude. In fig. 4, let HO be 
the horizon, EQ the equator, PP the 
axis of the earth, ZN the prime ver- 
tical, and ZL a parallel of latitude of 
any place Z. Then ZE is the lati- 


tude, (Art. 31,) and ZP the complement of the latitude ; but Zn 
the radius of the parallel of latitude ZL, is the sine of ZP, and 
therefore the cosine of the latitude. 

39. The Tropics are the parallels of latitude that pass through 
the solstices. The northern tropic is called the tropic of Cancer ; 
the southern, the tropic of Capricorn. 

40. The Polar Circles are the parallels of latitude that pass 
through the poles of the ecliptic, at the distance of 23i degrees 
from the pole of the earth. (Art. 33.) 

41. The earth is divided into five zones. That portion of the 
earth which lies between the tropics, is called the Torrid Zone ; 
that between the tropics and polar circles, the Temperate Zones ; 
and that between the polar circles and the poles, the Frigid 

42. The Zodiac is the part of the celestial sphere which lies 
about 8 degrees on each side of the ecliptic. This portion of the 
heavens is thus marked off by itself, because the planets are never 
seen further from the ecliptic than this limit. 

43. The elevation of the pole is equal to the latitude of the place. 
The arc PE (Fig. 4.)=ZO.\PO=ZE which equals the lati- 

44. The elevation of the equator is equal to the complement of 
the latitude. 

ZH=90. But ZE=Lat. /. EH=90 Lat.=colatitude. 

45. The distance of any place from the pole (or the polar dis* 
tance) equals the complement of the latitude. 

EP=90. But EZ=Lat. .-. ZP=90-Lat.=colatitude. 





46. THE apparent diurnal revolution of the heavenly bodies 
from east to west, is owing to the actual revolution of the earth 
on its own axis from west to east. If we conceive of a radius 6T 
the earth's equator extended until it meets the concave sphere of 
the heavens, then as the earth revolves, the extremity of this line 
would trace out a curve on the face of the sky, namely, the celes- 
tial equator. In curves parallel to this, called the circles of diurnal 
revolution, the heavenly bodies actually appear to move, every star 
having its own peculiar circle. After the learner has first rendered 
familiar the real motions of the earth from west to east, he may then, 
without danger of misconception, adopt the common language, 
that all the heavenly bodies revolve around the earth once a day 
from east to west, in circles parallel to the equator and to each other. 

47. The time occupied by a star in passing from any point in 
the meridian until it comes round to the same point again, is called 
a sidereal day, and measures the period of the earth's revolution 
on its axis. If we watch the returns of the same star from day to 
day, we shall find the intervals exactly equal to one another; 
that is, the sidereal days are all equal.* Whatever star we select 
for the observation, the same result will be obtained. The stars, 
therefore, always keep the same relative position, and have a 
common movement round the earth, a consequence that natu- 
rally flows from the hypothesis, that their apparent motion is all 
produced by a single real motion, namely, that of the earth. The 
sun, moon, and planets, revolve in like manner, but their returns to 
the meridian are not, like those of the fixed stars, at exactly equal 

48. The appearances of the diurnal motions of the heavenly 

* Allowance is here supposed to be made for the effects of precession, &c. 



bodies are different in different parts of the earth, since eveiy 
place has its own horizon, (Art. 15,) and different horizons are 
variously inclined to each other. Let us suppose the spectator 
viewing the diurnal revolutions, successively, from several different 
positions on the earth. 

49. If he is on the equator, his horizon passes through both poles ; 
for the horizon cuts the celestial vault at 90 degrees in every di- 
rection from the zenith of the spectator ; but the pole is likewise 
90 degrees from his zenith, and consequently, the pole must be 
in his horizon. The celestial equator coincides with his Prime 
Vertical, being a great circle passing through the east and 
west points. Since all the diurnal circles are parallel to the equa- 
tor, they are all, like the equator, perpendicular to his horizon. 
Such a view of the heavenly bodies, is called a right sphere ; or, 

A RIGHT SPHERE is one in which all the daily revolutions of 
the heavenly bodies are in circles perpendicular to the horizon. 

A right sphere is seen only at the equator. Any star situated 
in the celestial equator, would appear to rise directly in the east, at 
noon to pass through the zenith of the spectator, and to set directly in 
the west ; in proportion as stars are at a greater distance from the 
equator towards the pole, they describe smaller and smaller circles, 
until, near the pole, their motion is hardly perceptible. In a right 
sphere every star remains an equal time above and below the hori-- 
zon ; and since the times of their revolutions are equal, the veloci- 
ties are as the lengths of the circles they describe. Consequently, 
as the stars are more remote from the equator towards the pole, 
their motions become slower, until, at -the pole, the north star ap- 
pears stationary. 

50. If the spectator advances one degree towards the north 
pole, his horizon reaches one degree beyond the pole of the earth, 
and cuts the starry sphere one degree below the pole of the heav- 
ens, or below the north star, if that be taken as the place of the 
pole. As he moves onward towards the pole, his horizon contin- 
ually reaches further and further beyond it, until when he comes 
to the pole of the earth, and under the pole of the heavens, his 
horizon reaches on all sides to the equator and coincides with it. 


Moreover, since all the circles of daily motion are parallel to the 
equator, they become, to the spectator at the pole, parallel to the 
horizon. This is what constitutes a parallel sphere. Or, 

A PARALLEL SPHERE is that in which all the circles of daily 
motion are parallel to the horizon. 

51. To render this view of the heavens familiar, the learner 
should follow round in his mind a number of separate stars, one 
near the horizon, one a few degrees above it, and a third near the 
zenith. To one who stood upon the north pole, the stars of the 
northern hemisphere would all be perpetually in view when not 
obscured by clouds or lost in the sun's light, and none of those of 
the southern hemisphere would ever be seen. The sun would 
be constantly above the horizon for six months in the year, and 
the remaining six constantly out of sight. That is, at the pole 
the days and nights are each six months long. The phenomena 
at the south pole are similar to those at the north. 

52. A perfect parallel sphere can never be seen except at one 
of the poles, a point which has never been actually reached by 
man; yet the British discovery ships penetrated within a few 
degrees of the north pole, and of course enjoyed the view of a 
sphere nearly parallel. 

53. As the circles of daily motion are parallel to the horizon of 
the pole, and perpendicular to that of the equator, so at all places 
between the two, the diurnal motions are oblique to the horizon. 
This aspect of the heavens constitutes an oblique sphere, which is 
thus defined : 

An OBLIQUE SPHERE is that in which the circles of daily mo- 
tion are oblique to the horizon. 

Suppose for example the spectator is at the latitude of fifty de- 
grees. His horizon reaches 50 beyond the pole of the earth, and 
gives the same apparent elevation to the pole of the heavens. It 
cuts the equator, and all the circles of daily motion, at an angle 
of 40, being always equal to the co-altitude of the pole. Thus, 
let HO (Fig. 5,) represent the horizon, EQ the equator, and 
PP' the axis of the earth. Also, //, mm, &c. parallels of latitude. 



Then the horizon of a spectator Fig. 5. 

at Z, in latitude 50 reaches to 
50 beyond the pole (Art. 50) ; 
and the angle ECH, is 40. As 
we advance still further north, 
the elevation of the diurnal cir- 
cles grows less and less, and 
consequently the motions of the 
heavenly bodies more and more 
oblique, until finally, at the pole, 
where the latitude is 90, the 
angle of elevation of the equator 
vanishes, and the horizon and equator coincide with each other, 
as before stated. 

54. The CIRCLE OF PERPETUAL APPARITION, is the boundary of 
that space around the elevated pole, where the stars never set. 
Its distance from the pole is equal to the latitude of the place. 
For, since the altitude of the pole is equal to the latitude, a star 
whose polar distance is just equal to the latitude, will when at its 
lowest point only just reach the horizon ; and all the stars nearer 
the pole than this will evidently not descend so far as the horizon. 

Thus, mm (Fig. 5,) is the circle of perpetual apparition, be- 
tween which and the north pole, the stars never set, and its dis- 
tance from the pole OP is evidently equal to the elevation of the 
pole, and of course to the latitude. 

55. In the opposite hemisphere, a similar part of the sphere 
adjacent to the depressed pole never rises. Hence, 

The CIRCLE OF PERPETUAL OCCULTATION, is the boundary of that 
space around the depressed pole, within which the stars never rise. 
Thus, m'm 1 (Fig. 5,) is the circle of perpetual occultation, be- 
tween which and the south pole, the stars never rise. 

56. In an oblique sphere, the horizon cuts the circles of daily 
motion unequally. Towards the elevated pole, more than half 
the circle is above the horizon, and a greater and greater portion 
as the distance from the equator is increased, until finally, within 


the circle of perpetual apparition, the whole circle is above the 
horizon. Just the opposite takes place in the hemisphere next 
the depressed pole. Accordingly, when the sun is in the equator, 
as the equator and horizon, like all other great circles of the 
sphere, bisect each other, the days and nights are equal all over 
the globe. But when the sun is north of the equator, our days 
become longer than our nights, but shorter when the sun is 
south of the equator. Moreover, the higher the latitude, the 
greater is the inequality in the lengths of the days and nights. 
All these points will be readily understood by inspecting figure 5 

57. Most of the phenomena of the diurnal revolution can be 
explained, either on the supposition that the celestial sphere actu- 
ally all turns around the earth once in 24 hours, or that this mo- 
tion of the heavens is merely apparent, arising from the revolu- 
tion of the earth on its axis in the opposite direction, a motion 
of which we are insensible, as we sometimes lose the conscious- 
ness of our own motion in a ship or a steamboat, and observe all 
external objects to be receding from us with a common motion. 
Proofs entirely conclusive and satisfactory, establish the fact, that 
it is the earth and not the celestial sphere that turns ; but these 
proofs are drawn from various sources, and the student is not pre- 
pared to appreciate their value, or even to understand some of 
them, until he has made considerable proficiency in the study of 
astronomy, and become familiar with a great variety of astronom- 
ical phenomena. To such a period of our course of instruction, 
we therefore postpone the discussion of the hypothesis of the 
earth's rotation on its axis. 

58. While we retain the same place on the earth, the diurnal 
revolution occasions no change in our horizon, but our horizon 
goes round as well as ourselves. Let us first take our station on 
the equator at sunrise ; our horizon now passes through both the 
poles, and through the sun, which we are to conceive of as at a 
great distance from the earth, and therefore as cut, not by the 
terrestrial but by the celestial horizon. As the earth turns, the 
horizon dips more and more below the sun, at the rate of 15 de- 
grees for every hour, and, as in the case of the polar star, (Art. 50,) 


the sun appears to rise at the same rate. In six hours, therefore, 
it is depressed 90 degrees below the sun, which brings us directly 
under the sun, which, for our present purpose, we may consider as 
having all the while maintained the same fixed position in space. 
The earth continues to turn, and in six hours more, it completely 
reverses the position of our horizon, so that the western part of 
the horizon which at sunrise was diametrically opposite to the 
sun now cuts the sun, and soon afterwards it rises above the level 
of the sun, and the sun sets. During the next twelve hours, the 
sun continues on the invisible side of the sphere, -until the hori- 
zon returns to the position from which it started, and a new day 

59. Let us next contemplate the similar phenomena at the poles. 
Here the horizon, coinciding as it does with the equator, would 
cut the sun through its center, and the sun would appear to re- 
volve along the surface of the sea, one half above and the other 
half below the horizon. This supposes the sun in its annual 
revolution to be at one of the equinoxes. When the sun is north 
of the equator, it revolves continually round in a path which, 
during a single revolution, appears parallel to the equator, and it 
is constantly day ; and when the sun is south of the equator, it is, 
for the same reason, continual night. 

60. We have endeavored to conceive of the manner in which 
the apparent diurnal movements of the sun are really produced at 
two stations, namely, in the right sphere, and in the parallel sphere. 
These two cases being clearly understood, there will be little dif- 
ficulty in applying a similar explanation to an oblique sphere 


61. Artificial globes are of two kinds, terrestrial and celestial. 
The first exhibits a miniature representation of the earth ; the 
second, of the visible heavens ; and both show the various circles 
by which the two spheres are respectively traversed. Since all 
globes are similar solid figures, a small globe, imagined to be sit- 
uated at the center of the earth or of the celestial vault, may rep- 



resent all the visible objects and artificial divisions of either sphere, 
and with great accuracy and just proportions, though on a scale 
greatly reduced. The study of artificial globes, therefore, cannot 
be too strongly recommended to the student of astronomy.* 

62. An artificial globe is encompassed from north to south by 
a strong brass ring to represent the meridian of the place. This 
ring is made fast to the two poles and thus supports the globe, 
while it is itself supported in a vertical position by means of a 
frame, the ring being usually let into a socket in which it may be 
easily slid, so as to give any required elevation to the pole. The 
brass meridian is graduated each way from the equator to the 
pole 90, to measure degrees of latitude or declination, according 
as the distance from the equator refers to a point on the earth or 
in the heavens. The horizon is represented by a broad zone, made 
broad for the convenience of carrying on it a circle of azimuth, an- 
other of amplitude, and a wide space on which are delineated the 
signs of the ecliptic, and the sun's place for every day in the year ; 
not because these points have any special connexion with the hori- 
zon, but because this broad surface furnishes a convenient place 
for recording them. 

63. Hour Circles are represented on the terrestrial globe by 
great circles drawn through the pole of the equator ; but, on the 
celestial globe, corresponding circles pass through the poles of the 
ecliptic, constituting circles of celestial latitude, (Art. 37,) while the 
brass meridian, being a secondary to the equinoctial, becomes an 
hour circle of any star which, by turning the globe, is brought un- 
der it. 

64. The Hour Index is a small circle described around the pole 
of the equator, on which are marked the hours of the day. As 
this circle turns along with the globe, it makes a complete revo- 
lution in the same time with the equator ; or, for any less period, 

* It were desirable, indeed, that every student of the science should have the ce?es- 
tial globe at least, constantly before him. One of a small size, as eight or nine inches, 
will answer the purpose, although globes of these dimensions cannot usually be relied 
on for nice measurements. 

24 ' THE EARTH. 

the same number of degrees of this circle and of the equator pass 
under the meridian. Hence the hour index measures arcs of 
right ascension. (Art. 37.) 

65. The Quadrant of Altitude is a flexible strip of brass, gradu- 
ated into ninety equal parts, corresponding in length to degrees 
on the globe, so that when applied to the globe and bent so as 
closely to fit its surface, it measures the angular distance between 
any two points. When the zero, or the point where the gradua- 
tion begins, is laid on the pole of any great circle, the 90th degree 
will reach to the circumference of that circle, and being therefore 
a great circle passing through the pole of another great circle, it 
becomes a secondary to the latter. (Art. 21.) Thus the quadrant 
of altitude may be used as a secondary to any great circle on the 
sphere ; but it is used chiefly as a secondary to the horizon, the 
point marked 90 being screwed fast to the pole of the horizon, 
that is, the zenith, and the other end, marked 0, being slid along 
between the surface of the sphere and the wooden horizon. It 
thus becomes a vertical circle, on which to measure the altitude 
of any star through which it passes, or from which to measure 
the azimuth of the star, which is the arc of the horizon intercept- 
ed between the meridian and the quadrant of altitude passing 
through the star, (Art. 27.) 

66. To rectify the globe for any place, the north pole must be 
elevated to the latitude of the place (Art. 43) ; then the equator 
and all the diurnal circles will have their due inclination in respect 
to the horizon ; and, on turning the globe, (the celestial globe west, 
and the terrestrial east,) every point on either globe will revolve as 
the same point does in nature ; and the relative situations of all 
places will be the same as on the respective native spheres. 


67. To find the Latitude and Longitude of a place : Turn the 
globe so as to bring the place to the brass meridian ; then the de- 
gree and minute on the meridian directly over the place will indi- 
cate its latitude, and the point of the equator under the meridian, 
will show its longitude. % 


Ex. What are the Latitude and Longitude of the city of New 

68. To find a place having its latitude and longitude given: Bring 
to the brass meridian the point of the equator corresponding to 
the longitude, and then at the degree of the meridian denoting the 
latitude, the place will be found. 

Ex. What place on the globe is in Latitude 39 N. and Longi- 
tude 77 W. ? 

69. To find the bearing and distance of two places: Rectify the 
globe for one of the places (Art. 66) ; screw the quadrant of alti- 
tude to the zenith,* and let it pass through the other place. Then 
the azimuth will give the bearing of the second place from the 
first, and the number of degrees on the quadrant of altitude, mul- 
tiplied by 691, (the number of miles in a degree,) will give the 
distance between the two places. 

Ex. What is the bearing of New Orleans from New York, and 
what is the distance between these places ? 

70. To determine the difference of time in different places ; 
Bring the place that lies eastward of the other to the meridian, 
and set the hour index at XII. Turn the globe eastward until 
the other place comes to the meridian, then the index will point 
to the hour required. 

Ex. When it is noon at New York, what time is it at London ? 

71. The hour being given at any place, to tell what hour it is in 
any other part of the world : Find the difference of time between 
the two places, (Art. 70,) and, if the place whose time is required 
is eastward of the other, add this difference to the given time, but 
if westward, subtract it. 

Ex. What time is it at Canton, in China, when it is 9 o'clock 
A.M. at New York? 

72. To find the antceci,-\ the periceci, J and the antipodes^ of any 
* The zenith will of course be the point of the meridian over the place. 

t aVTl OtKOS. | ttfl (HKOS. CVTt Mf. 



place : Bring the given place to the meridian ; then, under the 
meridian, in the opposite hemisphere, in the same degree of lati- 
tude, will be found the antoeci. The same place remaining under 
the meridian, set the index to XII, and turn the globe until the 
other XII is under the index ; then the perioeci will be on the me- 
ridian, under the same degree of latitude with the given place, 
and the antipodes will be under the meridian, in the same latitude, 
in the opposite hemisphere. 

Ex. Find the antoeci, the perioeci, and the antipodes of the citi- 
zens^ *New York. 

.The aixtG&ci lifcve the same hour of the day, but different seasons 
of the year ; the perioeci have the same seasons, but opposite hours ; 
and the antipodes have both opposite hours and opposite seasons. 

73. To ratify the globe for the surfs place : On the wooden 
horizon, find the day of the month, and against it is given the sun's 
place in the ecliptic, expressed by signs and degrees.* Look for 
the same sign and degree on the ecliptic, bring that point to the 
meridian and set the hour index to XII. To all places under the 
meridian it will then be noon. 

Ex. Rectify the globe for the sun's place on the 1st of September. 

74. The latitude of the place being given, to find the time of the 
surfs rising and setting on any given day at that place : Having 
rectified the globe for the latitude, (Art. 66,) bring the sun's place 
in the ecliptic to the graduated edge of the meridian, and set the 
hour index to XII ; then turn the globe so as to bring the sun to 
the eastern and then to the western horizon, and the hour index 
will show the times of rising and setting respectively. 

Ex. At what time will the sun rise and set at New Haven, 
Lat. 41 18', on the 10th of July ? 


75. To find the Declination and Right Ascension of a heavenly 
body : Bring the place of the body (whether the sun or a star) to 
the meridian. Then the degree and minute standing over it will 

* The larger globes have the day of the month marked against the corresponding 
sign on the ecliptic itself. 


show its declination, and the point of the equinoctial under the 
meridian will give its right ascension. It will be remarked, that 
the declination and right ascension are found in the same manner 
as latitude and longitude on the terrestrial globe. Right ascen- 
sion is expressed either in degrees or in hours ; both being reck- 
oned from the vernal equinox, (Art. 37.) 

Ex. What is the declination and right ascension of the bright 
star Lyra ? also of the sun on the 5th of June ? 

76. To represent the appearance of the heavens . .^ . 
Rectify the globe for the latitude, bring the smfr P&ce, in 
ecliptic to the meridian, and set the hour index to. XII j^hpri ti 
the globe westward until the index points to t|i given hour, arid 
the constellations would then have the same appearance to an eye* 
situated at the center of the globe, as they have at . that moment 
in the sky. 

Ex. Required the aspect of the stars at New Haven, Lat. 41 
18', at 10 o'clock, on the evening of December 5th. 

77. To find the altitude and azimuth of any star : Rectify the 
globe for the latitude, and let the quadrant of altitude be screwed 
to the zenith, and be made to pass through the star. The arc on 
the quadrant, from the horizon to the star, will denote its altitude, 
and the arc of the horizon from the meridian to the quadrant, will 
be its azimuth. 

Ex. What are the altitude and azimuth of Sinus (the brightest 
of the fixed stars) on the 25th of December at 10 o'clock in the 
evening, in Lat. 41 ? 

78. To find the angular distance of two stars from each other . 
Apply the zero mark of the quadrant of altitude to one of the 
stars, and the point of the quadrant which falls on the other star, 
will show the angular distance between the two. 

Ex. What is the distance between the two largest stars of the 
Great Bear?* 

* These two stars are sometimes called " the Pointers," from the line which passes 
through them being always nearly in the direction of the north star. The angular 
distance between them is about 5, and may be learned as a standard for reference m 
estimating, by the eye, the distance between any two points on the celestial vault. 



79. To find the surfs meridian altitude, the latitude and day 
of the month being given: Having rectified the globe for the 
latitude, (Art. 66,) bring the sun's place in the ecliptic to the me- 
ridian, and count the number of degrees and minutes between 
that point of the meridian and the zenith. The complement of 
this arc will be the sun's meridian altitude. 

Ex. What is the sun's meridian altitude at noon on the 1st of 
August, in Lat. 41 18'? 



80. PARALLAX is the apparent change of place which bodies 
undergo by being viewed from different points. Thus in figure 
6, let A represent the earth, CH' the horizon, H'Z a quadrant of 

Fig. 6. 

a great circle of the heavens, extending from the horizon to the 
zenith ; and let E, F, G, H, be successive positions of the moon 
at different elevations, from the horizon to the meridian. Now a 
spectator on the surface of the earth at A, would refer the place 
of E to h, whereas, if seen from the center of the earth, it would 


appear at H'. The arc H'h is called the parallactic arc, and the 
angle H'E, or its equal AEC, is the angle of -parallax. The 
same is true of the angles at F, G, and H, respectively. 

81. Since then a heavenly body is liable to be referred to dif- 
ferent points on the celestial vault, when seen from different parts 
of the earth, and thus some confusion occasioned in the deter- 
mination of points on the celestial sphere, astronomers have agreed 
to consider the true place of a celestial object to be that where it 
would appear if seen from the center of the earth. The doctrine 
of parallax teaches how to reduce observations made at any place 
on the surface of the earth, to such as they would be if made 
from the center. 

82. The angle AEC is called the horizontal parallax, which 
may be thus defined. Horizontal Parallax, is the change of po- 
sition which a celestial body, appearing in the horizon as seen 
from the surface of the earth, would assume if viewed from the 
earth's center. It is the angle subtended by the semi-diameter 
of the earth, as viewed from the body itself. If we consider any 
one of the triangles represented in the figure, ACG for example, 

Sin. AGC : Sin. GAZ : : AC : CG 

...Sin. ParaDa^ 8 


Hence the sine of the angle of parallax, or (since the angle of 
parallax is always very small*) the parallax itself varies as the 
sine of the zenith distance of the body directly, and the distance 
of the body from the center of the earth inversely. Parallax, there- 
fore, increases as a body approaches the horizon, (but increasing 
with the sines, it increases much slower than in the simple ratio 
of the distance from the zenith,) and diminishes, as the distance 
from the spectator increases. Again, since the parallax AGC is as 
the sine of the zenith distance, let P represent the horizontal par- 
allax, and P' the parallax at any altitude ; then, 

* The moon, on account of its nearness to the earth, has the greatest horizontal 
parallax of any of the heavenly bodies ; yet this is less than 1 (being 57*) while the 
greatest parallax of any of the planets does not exceed 30". The difference in an 
arc of 1, between the length of the arc and the sine, is only O."18. 



P' : P::sin. zenith dist.: sin. 90V.P: 

sin. zen. dist. 

Hence, the horizontal parallax of a body equals its parallax at 
any altitude, divided by the sine of its distance from the zenith. 

83. From observations, therefore, on the parallax of a body at 
any elevation, we are enabled to find the angle subtended by the 
semi-diameter of the earth as seen from the body. Or ? if the 
horizontal parallax is given, the parallax at any altitude may be 
found, for 

P'=Pxsin. zenith distance. 

Hence, in the zenith the parallax is nothing, and is at its max- 
imum in the horizon. 

84. It is evident from the figure, that the effect of parallax 
upon the place of a celestial body is to depress it. Thus, in con- 
sequence of parallax, E is depressed by the arc H'^ ; F by the 
arc P/> ; G by the arc Rr ; while H sustains no change. Hence, 
in all calculations respecting the altitude of the sun, moon, or plan- 
ets, the amount of parallax is to be added ; the stars, as we shall 
see hereafter, have no sensible parallax. As the depression which 
arises from parallax is in the direction of a vertical circle, a body, 
when on the meridian, has only a parallax in declination ; but 
in other situations, there is at the same time a parallax in 
declination and right ascension ; for its direction being oblique 
to the equinoctial, it can be resolved into two parts, one of which 
(the declination) is perpendicular, and the other (the right ascen- 
sion) is parallel to the equinoctial. 

85. The mode of determining the horizontal parallax, is as 
follows : 

Let O, O', (Fig. 7,) be two places on the earth, situated under 
the same meridian, at a great distance from each other ; one place, 
for example, at the Cape of Good Hope, and the other in the north 
of Europe. The latitude of each place being known, the arc of 
the meridian OO' is known, and the angle OCO' also is known. 
Let the celestial body M, (the moon for example,) he observed 
simultaneously at O and O', and its zenith distance at each place 



accurately taken, namely, ZY and 

Z'Y' ; then the angles ZOM and 

Z'O'M, and of course their sup- 

plements COM,CO'M are found. 

Then in the quadrilateral figure 

COMO', we have all the angles 

and the two radii, CO, CO 7 , 

whence by joining OO', the side 

OM may be easily found. Hav- 

ing CO and OM, we may find 

CMO=sine of the angle of par- 

allax ; or (since the arc is very 

small) equals the parallax P'. 

But when M as seen from O is in the horizon, ZOM becomes a 

right angle, and its sine equal to radius. Then, CM being found, 

CM : CO : : 1 : P=horizontal parallax=. 


On this principle, the horizontal parallax of the moon was de- 
termined by La Caille and La Lande, two French astronomers, 
one stationed at the Cape of Good Hope, the other at Berlin ; and 
in a similar way the parallax of Mars was ascertained, by ob- 
servations made simultaneously at the Cape of Good Hope and 
at Stockholm. 

86. On account of the great distance of the sun, his horizontal 
parallax, which is only 8".6, cannot be accurately ascertained by 
this method. It can, however, be determined by means of the 
transits of Venus, a process to be described hereafter. 

87. The determination of the horizontal parallax of a celestial 
body is an element of great importance, since it furnishes the 
means of estimating the distance of the body from the center of 
the earth. Thus, if the angle AEC (Fig. 6,) be found, the radius 
of the earth AC being known, we have in the triangle AEC, 
right angled at A, the side AC and all the angles, to find the hypo- 
thenuse CE, which is the distance of the moon from 'he center 
of the earth. 



88. While parallax depresses the celestial bodies subject to it, 
refraction elevates them; and it affects alike the most distant 
as well as nearer bodies, being occasioned by the change of di- 
rection which light undergoes in passing through the atmos- 
phere. Let us conceive of the atmosphere as made up of a great 
number of concentric strata, as AA, BB, CC, and DD, (Fig. 8,) 

Fig. 8. 

increasing rapidly in density (as is known to be the fact) in ap- 
proaching near to the surface of the earth. Let S be a star, from 
which a ray of light S enters the atmosphere at #, where, being 
turned towards the radius of the convex surface, it would change 
its direction into the line ab, and again into be, and cO, reach- 
ing the eye at O. Now, since an object always appears in the 
direction in which the light finally strikes the eye, the star would 
be seen in the direction of the last ray cO, and the star would 
apparently change its place, in consequence of refraction, from 
S to S', being elevated out of its true position. Moreover, 
since on account of the constant increase of density in descend- 
ing through the atmosphere, the light would be continually turned 
out of its course more and more, it would therefore move, not 
in the polygon represented in the figure, but in a corresponding 
curve, whose curvature is rapidly increased near the surface of 
the earth. 

89. When a body is in the zenith, since a ray of light from it 
enters the atmosphere at right angles to the refracting medium, it 
suffers no refraction. Consequently, the position of the heavenly 


bodies, when in the zenith, is not changed by refraction, while, 
near the horizon, where a ray of light strikes the medium very 
obliquely, and traverses the atmosphere through its densest part, 
the refraction is greatest. The following numbers, taken at dif- 
ferent altitudes, will show how rapidly refraction diminishes from 
the horizon upwards. The amount of refraction at the horizon 
is 34- 00". At different elevations it is as follows. 






32' 00" 


1' 40" 


30 00 


1 09 

1 00 

24 25 



5 00 

10 00 



10 00 

5 20 



20 00 

2 39 



From this table it appears, that while refraction at the horizon 
is 34 minutes, at so small an elevation as only 10 minutes above 
the horizon it loses 2 minutes, more than the entire change from 
the elevation of 30 to the zenith. From the horizon to 1 above, 
the refraction is diminished nearly 10 minutes. The amount at 
the horizon, at 45, and at 90, respectively, is 34', 58", and 0. In 
finding the altitude of a heavenly body, the effect of parallax must 
be added, but that of refraction subtracted. 

90. Let us now learn the method, by which the amount of re- 
fraction at different elevations is ascertained. To take the sim- 
plest case, we will suppose ourselves in a high latitude, where 
some of the stars within the circle of perpetual apparition pass 
through the zenith of the place. We measure the distance of 
such a star from the pole when on the meridian above the pole, 
that is, in the zenith, where it is not at all affected by refraction, 
and again its distance from the pole in its lower culmination. 
Were it not for refraction, these two polar distances would be 
equal, since, in the diurnal revolution of a star, it is in fact always 
at the same distance from the pole ; but, on account of refraction, 
the lower distance will be less than the upper, and the difference 
between the two will show the amount of refraction at the lower 
culmination, the latitude of the place being known. 

Example. At Paris, latitude 48 50', a star was observed to 



pass the meridian & north of the zenith, and consequently, 41 4' 
from the pole.* It should have passed the meridian at the same 
distance below the pole, but the distance was found to be only 
40 57' 35". Hence, 41 4'-40 57' 35"=6' 25" is the refraction 
due to that altitude, that is, at the altitude of 7 46'=(48 50'- 
41 4'). By taking similar observations in various places situated 
in high latitudes, the amount of refraction might be ascertained 
for a number of different altitudes, and thus the law of increase 
in refraction as we proceed from the zenith towards the horizon, 
might be ascertained. 

91. Another method of finding the refraction at different alti- 
tudes, is as follows. Take the altitude of the sun or a star, whose 
right ascension and declination are known, and note the time by 
the clock. Observe also when it crosses the meridian, and the 
difference of time between the two observations will give the hour 
angle ZPx, (Fig. 9.) In this triangle ZPx we also know PZ the 

Fig. 9. 

co-latitude and Pa? the co-declination. Hence we can find the co 
altitude Zx, and of course the true altitude. Compare the alti- 
tude thus found with that before determined by observation, and 
the difference will be the refraction due to the apparent altitude. 

*For the polar distance of the place=90-48 50'=4P 10'; and 41<> 10'-6'= 


Ex. On May 1, 1738, at 5h. 20m. in the morning, Cussini ob- 
served the altitude of the sun's center at Paris to be 5 0' 14". The 
latitude of Paris being 48 50' 10", and the sun's declination at 
that time being 15 0' 25" : Required the refraction. 

By spherical trigonometry, Zx will be found equal to 85 10' 
8"; consequently, the true altitude was 4 49' 52". Now to 5 
0' 14", the apparent altitude, 9" must be added for parallax, 
and we have 5 0' 23" the apparent altitude corrected for parallax. 
Hence, 5 0' 23"-4 49' 52"=10' 31", the refraction at the ap- 
parent altitude 5 0' 14".* 

92. By these and similar methods, we could easily determine 
the refraction due to any elevation above the horizon, provided 
the refracting medium (the atmosphere) were always uniform. 
But this is not the fact : the refracting power of the atmosphere 
is altered by changes in density and temperature. f Hence in 
delicate observations, it is necessary to take into the account the 
state of the barometer and of the thermometer, the influence of 
the variations of each having been very carefully investigated, 
and rules having been given accordingly. With every precaution 
to insure accuracy, on account of the variable character of the 
refracting medium, the tables are not considered as entirely accu- 
rate to a greater distance from the zenith than 74 ; but almost all 
astronomical observations are made at a greater altitude than this. 

93. Since the whole amount of refraction near the horizon ex- 
ceeds 33', and the diameters of the sun and moon are severally 
less than this, these luminaries are in view both before they have 
actually risen and after they have set. 

94. The rapid increase of refraction near the horizon, is strik- 
ingly evinced by the oval figure which the sun assumes when 
near the horizon, and which is seen to the greatest advantage 
when light clouds enable us to view the solar disk. Were all , 

* Gregory's Ast. p. 65. 

t It is said that the effects of humidity are insensible ; for the most accurate 
experiments seem to prove that watery vapor diminishes the density of air in the 
same ratio as its own refractive power is greater than that- of air. (New Encyc. 
Brit. Ill, 762.) 


parts of the sun equally raised by refraction, there would be no 
change of figure ; but since the lower side is more refracted than 
the upper, the effect is to shorten the vertical diameter and thus 
to give the disk an oval form. This effect is particularly remark- 
able when the sun, at his rising or setting, is observed from the 
top of a mountain, or at an elevation near the sea shore ; for in 
such situations the rays of light make a greater angle than or- 
dinary with a perpendicular to the refracting medium, and the 
amount of refraction is proportionally greater. In some cases of 
this kind, the shortening of the vertical diameter of the sun has 
been observed to amount to 6', or about one fifth of the whole.* 

95. The apparent enlargement of the sun and moon in the hori- 
zon, arises from an optical illusion. These bodies in fact are 
not seen under so great an angle when in the horizon, as when on 
the meridian, for they are nearer to us in the latter case than in 
the former. The distance of the sun is indeed so great that it 
makes very little difference in his apparent diameter, whether he 
is viewed in the horizon or on the meridian ; but with the moon 
the case is otherwise ; its angular diameter, when measured with 
instruments, is perceptibly larger at the time of its culmination. 
Why then do the sun and moon appear so much larger when near 
the horizon ? It is owing to that general law, explained in optics, 
by which we judge of the magnitudes of distant objects, not 
merely by the angle they subtend at the eye, but also by our im- 
pressions respecting their distance, allowing, under a given angle, 
a greater magnitude as we imagine the distance of a body to be 
greater. Now, on account of the numerous objects usually in 
sight between us and the sun, when on the horizon, he appears 
much further removed from us than when on the meridian, and 
we assign to him a proportionally greater magnitude. If we view 
the sun, in the two positions, through smoked glass, no such dif- 
ference of size is observed, for here no objects are seen but the 
sun himself. 

* In extreme cold weather, this shortening of the sun's vertical diameter sometimes 
exceeds this amount. 



96. Twilight also is another phenomenon depending upon the 
agency of the earth's atmosphere. It is due partly to refraction 
and partly to reflexion, but mostly the latter. While the sun 
is within 18 of the horizon, before it rises or after it sets, some 
portion of its light is conveyed to us by means of numerous re- 
flections from the atmosphere. Let AB (Fig. 10,) be the horizon 

Fig. 10. 

of the spectator at A, and let SS be a ray of light from the sun 
when it is. two or three degrees below the horizon. Then to 
the observer at A, the segment of the atmosphere ABS would be 
illuminated. To a spectator at C, whose horizon was CD, the 
small segment SDx would be the twilight ; while, at E, the twi- 
light would disappear altogether. 

97. At the equator, where the circles of daily motion are per- 
pendicular to the horizon, the sun descends through 18 in an 
hour and twelve minutes (}f =l}h.), and the light of day there- 
fore declines rapidly, and as rapidly advances after daybreak in the 
morning. At the pole, a constant twilight is enjoyed while the sun 
is within 18 of the horizon, occupying nearly two thirds of the 
half year when the direct light of the sun is withdrawn, so that 
the progress from continual day to constant night is exceedingly 
gradual. To the inhabitants of an oblique sphere, the twilight 
is longer in proportion as the place is nearer the elevated pole. 

98. Were it not for the power the atmosphere has of dispersing 


the solar light, and scattering it in various directions, no objects 
would be visible to us out of direct sunshine ; every shadow of a 
passing cloud would be pitchy darkness ; the stars would be visi- 
ble all day, and every apartment into which the sun had not di- 
rect admission, would be involved in the obscurity of night. This 
scattering action of the atmosphere on the solar light, is greatly 
increased by the irregularity of temperature caused by the sun, 
which throws the atmosphere into a constant state of undulation, 
and by thus bringing together masses of air of different tempera- 
tures, produces partial reflections and refractions at their common 
boundaries, by which means much light is turned aside from the 
direct course, and diverted to the purposes of general illumination.* 
In the upper regions of the atmosphere, as on the tops of very 
high mountains, where the air is too much rarefied to reflect much 
light, the sky assumes a black appearance, and stars become visi- 
ble in the day time. 



99. TIME is a measured portion of indefinite duration. 

Any event may be taken as a measure of time, which divides 
a portion of duration into equal parts ; as the pulsations of the 
wrist, the vibrations of a pendulum, or the passage of sand from 
one vessel into another, as in the hour-glass. 

100. The great standard of time is the period of the revolution 
of the earth on its axis, which, by the most exact observations, is 
found to be always the same. The time of the earth's revolution 
on its axis is called a sidereal day, and is determined by the revo- 
lution of a star from the instant it crosses the meridian, until it 
comes round to the meridian again. This interval being called a 

* Herschel. 

TIME. 39 

sidereal day* it is divided into 24 sidereal hours. Observations 
taken upon numerous stars, in different ages of the world, show 
that they all perform their diurnal revolutions in the same time, 
and that their motion during any part of the revolution is per- 
fectly uniform. 

101. Solar time is reckoned by the apparent revolution of the 
sun, from the mefldian round to the same meridian again. Were 
the sun stationary in the heavens, like a fixed star, the time of its 
apparent revolution would be equal to the revolution of the earth 
on its axis, and the solar and the sidereal days would be equal. 
But since the sun passes from west to east, through 360 in 365^ 
days, it moves eastward nearly 1 a day, (59' 8". 3). While, 
therefore, the earth is turning round on its axis, the sun is moving 
in the same direction, so that when we have come round under 
the same celestial meridian from which we started, we do not 
find the sun there, but he has moved eastward nearly a degree, 
and the earth must perform so much more than one complete 
revolution, in order to come under the sun again. Now since a 
place on the earth gains 359 in 24 hours, how long will it take 
to gain 1 1 


359 : 24 : : 1 : =4 m nearly. 

Hence the solar day is about 4 minutes longer than the sidereal; 
and if we were to reckon the sidereal day 24 hours, we should 
reckon the solar day 24h. 4m. To suit the purposes of society at 
large, however, it is found most convenient to reckon the solar day 
24 hours, and to throw the fraction into the sidereal day. Then, 

24h. 4m. : 24 : : 24 : 23h. 56m. (23h. 56 m 4 S .09) = the length 
of a sidereal day. 

102. The solar days, however, do not always differ from the 
sidereal by precisely the same fraction, since the increments of 
right ascension, (Art. 37,) which measure the difference between 
a sidereal and a solar day, are not equal to each other. Apparent 
time, is time reckoned by the revolutions of the sun from tne 
meridian to the meridian again. These intervals being unequal 
of course the apparent solar days are unequal to each other. 


103. Mean time, is time reckoned by the average length of all 
the solar days throughout the year. This is the period which con- 
stitutes the civil day of 24 hours, beginning when the sun is on 
the lower meridian, namely, at 12 o'clock at night, and counted 
by 12 hours from the lower to the upper culmination, and from 
the upper to the lower. The astronomical day is the apparent 
solar day counted through the whole 24 hours, Listead of by pe- 
riods of 12 hours each, and begins at noon. Tnus 10 days and 
14 hours of astronomical time, would be 1 1 days and 2 hours of 
civil time. 

104. Clocks are usually regulated so as to indicate mean solar 
time ; yet as this is an artificial period, not marked off, like the 
sidereal day, by any natural event, it is necessary to know how 
much is to be added to or subtracted from the apparent solar 
time, in order to give the corresponding mean time. The inter- 
val by which apparent time diners from mean time, is called the 
equation of time. If a clock were constructed (as it may be) so 
as to keep exactly with the sun, going faster or slower according 
as the increments of right ascension were greater or smaller, and 
another clock were regulated to mean time, then the difference 
of the two clocks, at any period, would be the equation of time 
for that moment. If the apparent clock were faster than the 
mean, then the equation of time must be subtracted ; but if the 
apparent clock were slower than the mean, then the equation of 
time must be added, to give the mean time. The two clocks 
would differ most about the 3d of November, when the apparent 
time is 16 m greater than the mean (16 m 17 8 ). But, since appa- 
rent time is sometimes greater and sometimes less than mean 
time, the two must obviously be sometimes equal to each other. 
This is in fact the case four times a year, namely, April 15th, 
June 15th, September 1st, and December 22d. These epochs, 
however, do not remain constant ; for, on account of the change 
in the position of the perihelion, or the point where the earth is 
nearest the sun, (which shifts its place from west to east about 
12" a year,) the period when the sun's motions are most rapid, as 
well as that when they are slowest, will occur at different parts of 
the year. The change is indeed exceedingly small in a single 



year ; but in the progress of ages, the time of year when the sun's 
motion in its orbit is most accelerated, will not be, as at present, on 
the first of January, but may fall on the first of March, June, or 
any other day of the year, and the amount of the equation of 
time is obviously affected by the sun's distance from its perihelion, 
since the sun moves most rapidly when nearest the perihelion, and 
slowest when furthest from that point. 

105. The inequality of the solar days depends on two causes, the 
unequal motion of the earth in its orbit, and the inclination of the 
equator to the ecliptic. 

First, on account of the eccentricity* of the earth's orbit, the 
earth actually moves faster from the autumnal to the vernal equi- 
nox, than from the vernal to the autumnal, the difference of the 
two periods being about eight days (7d. 17h. 17m.) Thus, let 

Fig. 11. 

AEB (Fig. 11,) represent the earth's orbit, S being the place ol 

* The exact figure of the earth's orbit will be more particularly shown hereafter. 
All that the student requires to know, in order to understand the present subject, 



the sun, A the perihelion, or nearest distance of the earth from 
the sun, B the aphelion, or greatest distance, and E, E', E'', posi- 
tions of the earth in different points of its orbit. The place of 
the earth among the signs is the part of the heavens to which it 
would be referred if seen from the sun ; and the place of the sun 
is the part of the heavens to which it is referred as seen from the 
earth. Thus, when the earth is at E, it is said to be in Aries ; 
and as it moves from E through E' to A, its path in the heavens 
is through Aries, Taurus, Gemini, &c. Meanwhile the sun takes 
its place successively in Libra, Scorpio, Sagittarius, &c. Now, 
as will be shown more fully hereafter, the earth moves faster 
when proceeding from Aries through its perihelion to Libra, than 
from Libra through its aphelion to Aries, and, consequently, de- 
scribes the half of its apparent orbit in the heavens, T, 55, =*, 
sooner than the half ==, V3, T. The line of the apsides, that is, 
the major axis of the ellipse, is so situated at present, that the 
perihelion is in the sign Cancer, nearly 100 (99 30' 5") from the 
vernal equinox. The earth passes through it about the first of 
January, and then its velocity is the greatest in the whole year, 
being always greater as the distance is less, the angular velocity 
being inversely as the square of the distance, as will be shown by 
and by. 

106. But differences of time are not reckoned on the eclip- 
tic, but on the equinoctial ; for the ecliptic being oblique to the 
meridian in the diurnal motion, and cutting it at different angles at 
different times, equal portions will not pass under the meridian in 
equal times, and therefore such portions could not be employed, as 
they are in the equinoctial, as measures of time. If therefore the 
sun moved uniformly in his orbit, so as to make the daily incre- 
ments of longitude equal, still the corresponding arcs of right as- 
cension, which determine the lengths of the solar day, would be 
unequal. Let us start from the equinox, from which both longi- 
tude and right ascension are reckoned, the former on the ecliptic, 

is that the earth's orbit is an ellipse, and that the earth's real motion, and conse- 
quently the sun's apparent motion, is greater in proportion as the earth is nearer 
the sun. 



the latter on the equinoctial. Suppose the sun has described 70 
of longitude ; then to ascertain tne corresoonding arc of right as- 
cension, we let a meridian pass through tne sun : the point where 
it cuts the equator gives the sun's right ascension. Now since the 
ecliptic makes an acute angle with the meridian, while the equi- 
noctial makes a right angle with it, consequently the arc of longi- 
tude is greater than the arc of right ascension. The difference, 
however, grows constantly less and less as we approach the tropic, 
as the angle made between the ecliptic and the meridian constantly 
increases, until, when we reach the tropic, the meridian is at right 
angles to both circles, and the longitude and right ascension each 
equals 90, and they are of course equal to each other. Beyond 
this, from the tropic to the other equinox, the arc of the ecliptic 
intercepted between the meridian and the autumnal equinox being 
greater than the corresponding arc of the equinoctial, of course 
its supplement, which measures the longitude, is less than the sup- 
plement of the corresponding arc of the equator which measures 
the right ascension. At the autumnal equinox again, the right 
ascension and longitude become equal. In a similar manner we 
might show that the daily increments of longitude and right as- 
cension are unequal. 

In order to illustrate the foregoing points, let T *** (Fig. 12,) 

Fig. 12. 

represent the equator, T T =*= the ecliptic, and PSE, PS'E', two 
meridians meeting the sun in S and S'. Then in the triangle TES, 


the arc of longitude TS, is greater than TE, the corresponding 
arc of right ascension; but towards the tropic the difference 
between the two arcs evidently grows less and less, until at T 
the arcs become equal, being each 90. But, beyond the tropic, 
since TE'===, TS'^=, are equal to each other, each being equal 
to 180, and since S'=^= is greater than E'=s=, therefore TS' must 
be less than TE'. 

107. As the whole arc of right ascension reckoned from the 
first of Aries, does not keep uniform pace with the corresponding 
arc of longitude, so the daily increments of right ascension differ 
from those of longitude. If we suppose in the quadrant TT, 
points taken to mark the progress of the sun from day to day, and 
let meridians like PSE pass through these points, the arc of the 
ecliptic embraced between the meridians will be the daily incre- 
ments of longitude, while the corresponding parts of the equinoc- 
tial will be the daily increments of right ascension. Near T, the 
oblique direction in which the ecliptic cuts the meridian, will make 
the daily increments of longitude exceed those of right ascension ; 
but this advantage is diminished as we approach the tropic, where 
the ecliptic becomes less oblique, and finally parallel to the equi- 
noctial ; while the convergence of the meridians contributes still 
farther to lessen the ratios of the daily increments of longitude to 
those of right ascension. Hence, at first, the diurnal arcs of 
right ascension are less than those of longitude, but afterwards 
greater ; and they continue greater for a similar distance beyond 
the tropic. 

108. From the foregoing considerations it appears, that the 
diurnal arcs of right ascension, by which the difference between 
the sidereal and the solar days is measured, are unequal, on ac- 
count both of the unequal motion of the sun in his orbit, and of 
the inclination of his orbit to the equinoctial. 

109. As astronomical time commences when the sun is on the 
meridian, so sidereal time commences when the vernal equinox 
is on the meridian, and is also counted from to 24 hours. By 
3 o'clock, for instance, of sidereal time, we mean that it is three 


hours since the vernal equinox crossed the meridian ; as we say it 
is 3 o'clock of astronomical or of civil time, when it is three hours 
since the sun crossed the meridian. 


110. The astronomical year is the time in which the sun makes 
one revolution in the ecliptic, and consists of 365d. 5h. 48m. 5P.60. 
The civil year consists of 365 days. The difference is nearly 6 
hours, making one day in four years. 

111. The most ancient nations determined the number of days 
in the year by means of the stylus, a perpendicular rod which 
cast its shadow on a smooth plane, bearing a meridian line. The 
time when the shadow was shortest, would indicate the day of 
the summer solstice ; and the number of days which elapsed until 
the shadow returned to the same length again, would show the 
number of days in the year. This was found to be 365 whole 
days, and accordingly this period was adopted for*the civil year. 
Such a difference, however, between the civil and astronomical 
years, at length threw all dates into confusion. For, if at first 
the summer solstice happened on the 21st of June, at the end of 
four years, the sun would not have reached the solstice until the 
22d of June, that is, it would have been behind its time. At the 
end of the next four years the solstice would fall on the 23d ; 
and in process of time it would fall successively on every day of 
the year. The same would be true of any other fixed date. 
Julius Caesar made the first correction of the calendar, by intro- 
ducing an intercalary day every fourth year, making February 
to consist of 29 instead of 28 days, and of course the whole year 
to consist of 366 days. This fourth year was denominated Bis- 
sextile.*, It is also called Leap Year. 

112. But the true correction was not 6 hours, but 5h. 49m.; 
hence the intercalation was too great by 1 1 minutes. This small 
fraction would amount in 100 years to of a day, and in 1000 

* The sextus dies ante Kalendas being reckoned twice, (Bis). 


years to more than 7 days. From the year 325 to 1582, it had 
in fact amounted to about 10 days; for it was known that in 325, 
the vernal equinox fell on the 21st of March, whereas, in 1582 it 
fell on the llth. In order to restore the equinox to the same date, 
Pope Gregory XIII decreed, that the year should be brought for- 
ward ten days, by reckoning the 5th of October the 15th. In or- 
der to prevent the calendar from falling into confusion afterwards, 
the following rule was adopted : 

Every year whose number is not divisible by 4 without a re- 
mainder, consists of 365 days ; every year which is so divisible, but 
is not divisible by 100, of 366 ; every year divisible by 100 but not 
by 400, again of 365 ; and every year divisible by 400, of 366. 

Thus the year 1838, not being divisible by four, contains 365 days, 
while 1836 and 1840 are leap years. Yet to make every fourth 
year consist of 366 days would increase it too much by about 
of a day in 100 years ; therefore every hundredth year has only 
365 days. Thus 1800, although divisible by 4, was not a leap 
year, but a common year. But we have allowed a whole day 
in a hundred Jrears, whereas we ought to have allowed only three 
fourths of a day. Hence, in 400 years we should allow a day too 
much, and therefore we let the 400th year remain a leap year. 
This rule involves an error of less than a day in 4237 years.* If 
the rule were extended by making every year divisible by 4,000 
(which would now consist of 366 days) to consist of 365 days, the 
error would not be more than one day in 100,000 years. f 

113. This reformation of the calendar was not adopted in Eng- 
land until 1752, by which time the error in the Julian calendar 
amounted to about 11 days. The year was brought forward, by 
reckoning the 3d of September the 14th. Previous to that time 
the year began the 25th of March ; but it was now made to be- 
gin on the 1st of January, thus shortening the preceding year, 
1751, one quarter. J 

* Woodhouse, p. 874. t Herscliel's Ast. p. 384. 

\ Russia, and the Greek Church generally, adhere to the old style. In order to make 
the Russian dates correspond to ours, we must add to them 12 days. France and other 
Catholic countries, adopted the Gregorian calendar soon after it was promulgated. 


114. As in the year 1582, the error in the Julian calendar 
amounted to 10 days, and increased by of a day in a century, 
at present the correction is 12 days ; and the number of the year 
will differ by one with respect to dates between the 1st of Janu- 
ary and the 25th of March. 

Examples. General Washington was born Feb. 11, 1731, old 
style ; to what date does this correspond in new style ? 

As the date is the earlier part of the 18th century, the correc- 
tion is 1 1 days, which makes the birth day fall on the 22d of 
February ; and since the year 1731 closed the 25th of March, 
while according to new style 1732 would have commenced on 
the preceding 1st of January ; therefore, the time required is Feb. 
22, 1732. It is usual, in such cases, to write both years, thus- 
Feb. 11,1731-2,0.8. 

2. A great eclipse of the sun happened May 15th, 1836 ; to 
what date would this time correspond in old style ? 

Ans. May. 3d, 

115. The common year begins and ends on the same day of 
the week ; but leap year ends one day later in the week than it began. 

For 52x7=364 days; if therefore the year begins on Tues- 
day, for example, 364 days would complete 52 weeks, and one 
day would be left to begin another week, and the following year 
would begin on Wednesday. Hence, any day of the month is one 
day later in the week than the corresponding day of the preceding 
year. Thus, if the 16th of November, 1838, falls on Friday, 
the 16th of November, 1837, fell on Thursday, and will fall in 
1839 on Saturday. But if leap year begins on Sunday, it ends 
on Monday, and the following year begins on Tuesday ; while 
any given day of the month is two days later in the week than 
the corresponding date of the preceding year. 

116. Fortunately for astronomy, the confusion of dates involved 
in different calendars affects recorded observations but little. Re- 
markable eclipses, for example, can be calculated back for several 
thousand years, without any danger of mistaking the day of their 
occurrence ; and whenever any such eclipse is so interwoven with 
the account given by an ancient author of s6me historical event, 


as to indicate precisely the interval of time between the eclipse 
and the event, and at the same time completely to identify the 
eclipse, that date is recovered and fixed forever.* 




117. THE most ancient astronomers employed no instruments 
for measuring angles, but acquired their knowledge of the heav- 
enly bodies by long continued and most attentive inspection with 
the naked eye. In the Alexandrian school, about 300 years before 
the Christian era, instruments began to be freely used, and thence- 
forward trigonometry lent a powerful aid to the science of astron- 
omy. Tycho Brahe, in the 16th century, formed a new era in 
poetical astronomy, and carried the measurement of angles to 
10 ', a degree of accuracy truly wonderful, considering that he 
had not the advantage of the telescope. By the application of 
the telescope to astronomical instruments, a far better defined view 
of objects was acquired, and a far greater degree of refinement 
was attainable. The astronomers royal of Great Britain perfected 
the art of observation, bringing the measurement of angles to 1", 
and the estimation of differences of time to T V of a second. Be- 
yond this degree of refinement it is supposed that we cannot 
advance, since unavoidable errors arising from the uncertainties 
of refraction, and the necessary imperfection of instruments, for- 
bid us to hope for a more accurate determination than this. But 
a little reflection will show us, that I" on the limb of an astro- 
nomical instrument, must be a space exceedingly small. Suppose 
the circle, on which the angle is measured, be one foot in diameter. 

* An elaborate view of the Calendar may be found in Delambre's Astronomy, t. III. 
A useful table for finding the day of the week of any given date, is inserted in the 
American Almanac for 1832, p. 72. 


Then - 12X3 ' 14159 = T y inch = space occupied by 1. Hence 

= =space of T, and ^-,-7;77= s P ace of 1". Such minute 

10x60 600 36000 

angles can be measured only by large circles. If, for example, 
a circle is 20 feet in diameter, a degree on its periphery would 
occupy a space 20 times as large as a degree on a circle of 1 foot. 
A degree therefore of the limb of such an instrument would 
occupy a space of 2 inches : one minute, gV of an inch ; and one 
second, TJ Vo- of an inch. 

118. But the actual divisions on the limb of an astronomical 
instrument never extend to seconds : in the smaller instruments 
they reach only to 10', and on the largest rarely lower than 1'. 
The subdivision of these spaces is carried on by means of the 
Vernier, which may be thus defined : 

A VERNIER is a contrivancce attached to the graduated limb of 
an instrument, for the purpose of measuring aliquot parts of the 
smallest spaces, into which the instrument is divided. 

The vernier is usually a narrow zone of metal, which is made 
to slide on the graduated limb. Its divisions correspond to those 
on the limb, except that they are a little larger,* one tenth, for 
example, so that ten divisions on the vernier would equal eleven 
on the limb. Suppose now that our instrument is graduated to 
degrees only, but the altitude of a certain star is found to be 40 
and a fraction, or 40 +x. In order to estimate the amount of this 
fraction, we bring the zero point of the vernier to coincide with 
the point which indicates the exact altitude, or 40 +x. We then 
look along the vernier until we find where one of its divisions 
coincides with one of the divisions of the limb. Let this be at the 
fourth division of the vernier. In four divisions, therefore, the ver- 
nier has gained upon the divisions of the limb, a space equal to x ; 
and since, in the case supposed, it gains T y of a degree, or 6' at each 
division, the entire gain is 24', and the arc in question is 40 24'. 

119. As the vernier is used in the barometer, where its applica- 

* In the more modern instruments the divisions of the vernier are smaller than those 
of the limb 




Fig. 13. 




tion is more easily seen than in astronomical instruments, while the 
principle is the same in both cases, let us 
see how it is applied to measure the ex- 
act height of a column of mercury. Let 
AB (Fig. 13,) represent the upper part 
of a barometer, the level of the mercury 
being at C, namely, at 30.3 inches, and 
nearly another tenth. The vernier being 
brought (by a screw which is usually at- 
tached to it) to coincide with the surface 
of the mercury, we look along down the 
scale, until we find that the coincidence 
is at the 8th division of the vernier. 
Now as the vernier gains T V of T V=T^o- 
of an inch at each division upward, it of 
course gains T J T in eight divisions. The fractional quantity, there- 
fore, is .08 of an inch, and the height of the mercury is 30.38. If 
the divisions of the vernier were such, that each gained g-V (when 
60 on the vernier would equal 61 on the limb) on a limb divided 
into degrees, we could at once take off minutes ; and were the limb 
graduated to minutes, we could in a similar way read off seconds. 

120. The instruments most used for astronomical observations, 
are the Transit Instrument, the Astronomical Clock, the Mural 
Circle, and the Sextant. A large portion of all the observations, 
made in an astronomical observatory, are taken on the meridian. 
When a heavenly body is on the meridian, being at its highest 
point above the horizon, it is then least affected by refraction and 
parallax ; its zenith distance (from which its altitude and decli- 
nation are easily derived) is readily estimated ; and its right as- 
cension may be very conveniently and accurately determined by 
means of the astronomical clock. Having the right ascension 
and declination of a heavenly body, various other particulars re- 
specting its position may be found, as we shall see hereafter, by 
the aid of spherical trigonometry. Let us then first turn our at- 
tention to the instruments employed for determining the right 
ascension and declination. They are the Transit Instrument, the 
Astronomical Clock, and the Mural Circle. 



121. The Transit Instrument is a telescope, which is fixed 
permanently in the meridian, and moves only in that plane. It 
rests on a horizontal axis, which consists of two hollow cones 
applied base to base, a form uniting lightness and strength. The 
two ends of the axis rest on two firm supports, as pillars of stone, 
for example, usually built up from the ground, and so related to 
the building as to be as free as possible from all agitation. In 
figure 14, AD represents the telescope, E, W, massive stone pillars 
supporting the horizontal axis, beneath which is seen a spirit level, 
(which is used to bring the axis to a horizontal position,) and n a 
vertical circle graduated into degrees and minutes. This circle 
serves the purpose of placing the instrument at any required alti- 
tude or distance from the zenith, and of course for determining 
altitudes and zenith distances. 

Fig. 14. 

122. Various methods are described in works on practical as- 
tronomy, for placing the Transit Instrument accurately in the 
meridian. The following method by observations on the pole 
star, may serve as an example If the instrument be directed 



towards the north star, and so adjusted that the star Alioth (the 
first in the tail of the Great Bear) and the pole star are both in 
the same vertical circle, the former below the pole and the latter 
above it, the instrument will be nearly in the plane of the meridian. 
To adjust it more exactly, compare the time occupied by the pole 
star in passing from its upper to its lower culmination, with the 
time of passing from its lower to its upper culmination. These 
two intervals ought to be precisely equal ; and if they are so, the 
iustrument is truly placed in the meridian ; but if they are not 
equal, the position of the instrument must be shifted until they 
become exactly equal. 

123. The line of collimation of a telescope, is a line joining the 
center of the object glass with the center of the eye glass. When 
the transit instrument is properly adjusted, this line, as the instru- 
ment is turned on its axis, moves in the plane of the meridian. 
Having, by means of the vertical circle n, set the instrument at 
the known altitude or zenith distance of any star, upon which we 
wish to make observations, we wait until the star enters the field 
of the telescope, and note the exact instant when it crosses the 
vertical wire in the center of the field, which wire marks the true 
plane of the meridian. Usually, however, there are placed in the 
focus of the eye glass five parallel wires or threads, two on each 
side of the central wire, and all 
at equal distances from each 
other, as is represented in the 
following diagram. The time 
of arriving at each of the wires 
being noted, and all the times 
added together and divided by 
the number of observations, the 
result gives the instant of cross- 
ing the central wire. 

124. The Astronomical Clock 
is the constant companion of the 
Transit Instrument. This clock is so regulated as to keep exact 
pace with the stars, and of course with the revolution of the earth 


on its axis ; that is, it is regulated to sidereal time. It measures 
the progress of a star, indicating an hour for every 15, and 24 
hours for the whole period of the revolution of the star. Sidereal 
time, it will be recollected, commences when the vernal equinox 
is on the meridian, just as solar time commences when the sun is 
on the meridian. Hence, the hour by the sidereal clock has no 
correspondence with the hour of the day, but simply indicates 
how long it is since the equinoctial point crossed the meridian. 
For example, the clock of an observatory points to 3h. 20m. ; this 
may be in the morning, at noon, or any other time of the day, since 
it merely shows that it is 3h. 20m. since the equinox was on the 
meridian. Hence, when a star is on the meridian, the clock 
itself shows its right ascension ; and the interval of time between 
the arrival of any two stars upon the meridian, is the measure of 
their difference of right ascension. 

125. Astronomical clocks are made of the best workmanship, 
with a compensation pendulum, and every other advantage which 
can promote their regularity. The Transit Instrument itself, 
when once accurately placed in the meridian, affords the means 
of testing the correctness of the clock, since one revolution of a 
star from the meridian to the meridian again, ought to correspond 
to exactly 24 hours by the clock, and to continue the same from 
day to day ; and the right ascension of various stars, as they cross 
the meridian, ought to be such by the clock as they are given in 
the tables, where they are stated according to the most accurate 
determinations of astronomers. Or by taking the difference of 
right ascension of any two stars on successive days, it will be seen 
whether the going of the clock is uniform for that part of the 
day ; and by taking the right ascension of different pairs of stars, 
we may learn the rate of the clock at various parts of the day. 
We thus learn, not only whether the clock accurately measures 
the length of the sidereal day, but also whether it goes uniformly 
from hour to hour. 

Although astronomical clocks have been brought to a great de- 
gree of perfection, so as to vary hardly a second for many months, 
yet none are absolutely perfect, and most are so far from it as to 
require to be corrected by means of the Transit Instrument every 


few days. Indeed, for the nicest observations, it is usual not t 
attempt to bring the clock to an absolute state of correctness, but 
after bringing it as near to such a state as can conveniently be 
done, to ascertain how much it gains or loses in a day ; that is, to 
ascertain its rate of going, and to make allowance accordingly. 

126. The vertical circle (n, Fig. 14,) usually connected with 
the Transit Instrument, affords the means of measuring arcs on 
the meridian, as meridian altitudes, zenith distances, and decli- 
nations ; but as the circle must necessarily be small, and there- 
fore incapable of measuring very minute angles, the Mural Cir- 
cle is usually employed for measuring arcs of the meridian. The 
Mural Circle is a graduated circle, usually of very large size, fixed 
permanently in the plane of the meridian, and attached firmly to 
a perpendicular wall. It is made of large size, sometimes 1 1 feet 
in diameter, in order that very small angles may be measured on 
its limb ; and it is attached to a massive wall of solid masonry in 
order to insure perfect steadiness, a point the more difficult to 
attain in proportion as the instrument is heavier. The annexed 
diagram represents a Mural Circle fixed to its wall and ready for 
observations. It will be seen that every expedient is employed 
to give the instrument firmness of parts and steadiness of position. 
Its radii are composed of hollow cones, uniting lightness and 
strength, and its telescope revolves on a large horizontal axis, 
fixed as firmly as possible in a solid wall. The graduations are 
made on the outer rim of the instrument, and are read off by six 
microscopes (called reading microscopes) attached to the wall, one 
of which is represented at A, and the places of the five others 
are marked by the letters B, C, D, E, F. Six are used, in order 
that by taking the mean of such a number of readings, a higher 
degree of accuracy may be insured, than could be obtained by a 
single reading. In a circle of six feet diameter, like that repre- 
sented in the figure, the divisions may be easily carried to five 
minutes each. The microscope (which is of the variety called 
compound microscope) forms an enlarged image of each of these 
divisions in the focus of the eye glass. With it is combined the 
principle of the micrometer. This is effected by placing in the 
focus a delicate wire, which may be moved by means of a screw 


Fig. 16. 


m a direction parallel to the divisions of the limb, and which is so 
adjusted to the screw as to move over the whole magnified space 
of five minutes by five revolutions of the screw. Of course one 
revolution of the screw measures one minute. Moreover, if the 
screw itself is made to carry an index attached to its axis and re- 
volving with it over a disk graduated into sixty equal parts, then 
the space measured by moving the index over one of these parts, 
will be one second. 

We have been thus minute in the description of this instrument, 
in order to give the learner some idea of the vast labor and great 
patience demanded of practical astronomers, in order to obtain 
measurements of such extreme accuracy as those to which they 

On account of the great dimensions of this circle, and the ex- 
pense attending it, as well as the difficulty of supporting it firmly, 
sometimes only one fourth of it is employed, constituting the Mu- 
ral Quadrant. This instrument has the disadvantage, however, 



of being applicable to only one hemisphere at a time, either the 
northern or the southern, according as it is fixed to the eastern 
or the western side of the wall. 

127. We have before shown (Art. 124,) the method of finding 
the right ascension of a star by means of the Transit Instrument 
and the clock. The declination may be obtained by means of the 
mural circle in several different ways, our object being always to 
find the distance of the star, when on the meridian, from the equa- 
tor (Art. 37.) First, the declination may be found from the me- 
ridian altitude. Let S (Fig. 17,) be the place of a star when 
on the meridian. Then its meridian altitude will be SH, which 
will best be found by taking its ze- 
nith distance ZS, of which it is the 

complement. From SH, subtract EH, 
the elevation of the equator, which 
equals the co-latitude of the place of 
observation, (Art. 44,) and the remain- 
der SE is the declination. Or if the 
star is nearer the horizon than the 
equator is, as at S', subtract its me- 
ridian altitude from the co-latitude, for 
the declination. Secondly, the declination may be found from 
the north polar distance, of which it is the complement. Thus 
from P to E is 90. Therefore, PE-PS=90-PS=SE=the 
decimation. The height of the pole P is always known when the 
latitude of the place is known, being equal to the latitude. 

128. The astronomical instruments already described are adapt- 
ed to taking observations on the meridian only ; but we some- 
times require to know the altitude of a celestial body when it is 
not on the meridian, and its azimuth, or distance from the meridian 
measured on the horizon ; and also the angular distance between 
two points on any part of the celestial sphere. An instrument 
especially designed to measure altitudes and azimuths, is called an 
Altitude and Azimuth Instrument, whatever may be its particular 
form. When a point is on the horizon its distance from the me- 
ridian, or its azimuth, may be taken by the common surveyor's 


compass, the direction of the meridian being determined by the 
needle ; but when the object, as a star, is not on the horizon, its 
azimuth, it must be remembered, is the arc of the horizon froir 
the meridian to a vertical circle passing through the star (Art. 27) ; 
at whatever different altitudes, therefore, two stars may be, and 
however the plane which passes through them may be inclined to 
the horizon, still it is their angular distance measured on the hori- 
zon which determines their difference of azimuth. Figure 18 rep- 
resents an Altitude and Azimuth Instrument, several of the usual 
appendages and subordinate contrivances being omitted for the 
sake of distinctness and simplicity. Here abc is the vertical or 
altitude circle, and EFG the horizontal or azimuth circle ; AB is a 

Fig. 18. 

telescope mounted on a horizontal axis and capable of two mo- 
tions, one in altitude parallel to the circle abc, and the other in 
azimuth parallel to EFG. Hence it can be easily brought to bear 
upon any object. At m, under the eye glass of the telescope, is a 
small mirror placed at an angle of 45 with the axis of the tele- 
scope, by means of which the image of the object is reflected up- 
wards, so as to be conveniently presented to the eye of the ob- 



server. At d is represented a tangent screw, by which a slow 
motion is given to the telescope at c. At h and g are seen two 
spirit levels at right angles to each other, which show when the 
azimuth circle is truly horizontal. The instrument is supported 
on a tripod, for the sake of greater steadiness, each foot being 
furnished with a screw for levelling. 

129. The sextant is one of the most useful instruments, both 
to the astronomer and the navigator, and will therefore merit 
particular attention. In figure 19, 1 and H are two small mirrors, 
and T a small telescope. I D represents a movable arm, or 
radius, which carries an index at D. The radius turns on a pivot 
at I, and the index moves on a graduated arc EF. I is called 

Fig. 19. 

the Index Glass and H the Horizon Glass. The under part only 
of the horizon glass is coated with quicksilver, the upper part 
being left transparent ; so that while one object is seen through 
the upper part by direct vision, another may be seen through 
the lower part by reflexion from the two mirrors. The instru- 
ment is so contrived, that when the index is moved up to F, 
where the zero point is placed, or the graduation begins, the two 


reflectors I and H are exactly parallel to each other. If w 
now look through the telescope, T, so pointed as to see the star 
S through the transparent part of the horizon glass, we shall 
see the same star, in the same place, reflected from the silvered 
part ; for the star (or any similar object) is at such a distance 
that the rays of light which strike upon the index glass I, are 
parallel to those which enter the eye directly, and will exhibit 
the object at the same place. Now, suppose we wish to meas- 
ure the angular distance between two bodies, as the moon and a 
star, and let the star be at S and the moon at M. The telescope 
being still directed to S, turn the index arm I D from F towards 
E until the image of the moon is brought down to S, its lower 
limb just touching S. By a principle in optics, the angular dis- 
tance which the image of the moon passes over, is twice that of 
the mirror I. But the mirror has passed over the graduated arc 
FD ; therefore double that arc is the angular distance between 
the star and the moon's lower limb. If we then bring the upper 
limb into contact with the star, the sum of both observations, 
divided by 2, will give the angular distance between the star 
and the moon's center. As each degree on the limb EF meas- 
ures two degrees of angular distance, hence the divisions for sin- 
gle degrees are in fact only half a degree asunder ; and a sextant, 
or the sixth part of the circle, measures an angular distance of 
120. The upper and lower points in the disk of the sun or of 
the moon, may be considered as two separate objects, whose 
distance from each other may be taken in a similar manner, 
and thus their apparent diameters at any time be determined. 
We may select our points of observation either in a vertical, or 
in a horizontal direction. 

130. If we make a star, or the limb of the sun or moon, one of 
the objects, and the point in the horizon directly beneath, the oth- 
er, we thus obtain the altitude of the object. In this observation, 
the horizon is viewed through the transparent part of the hori- 
zon glass. At sea, where the horizon is usually well defined, the 
horizon itself may be used for taking altitudes ; but on land, in- 
equalities of the earth's surface, oblige us to have recourse to an 
artificial horizon. This, in its simple state, is a basin of either 


water or quicksilver. By this means we see the image of the 
sun (or other body) just as far below the horizon as it is in reality 
above it. Hence, if we turn the index glass until the limb of the 
sun, as reflected from it, is brought into contact with the image 
seen in the artificial horizon, we obtain double the altitude.* 

The sextant must be held in such a manner, that its plane shall 
pass through the plane of the two objects. It must be held 
therefore in a vertical plane in taking altitudes, and in a horizontal 
plane in taking the horizontal diameters of the sun and moon. 
Holding the instrument in the true plane of the two bodies, whose 
angular distance is measured, is indeed the most difficult part of 
the operation. 

The peculiar value of the sextant consists in this, that the ob- 
servations taken with it are not affected by any motion in the 
observer ; hence it is the chief instrument used for angular meas- 
urements at sea. 

131. Examples illustrating the use of the Sextant. 
Ex. 1. Alt. 0's lower limb, . . 49 10' 00" 
's semi-diameter, . . , 15 51 

49 25' 51" 
Subtract Refraction, . . 00 00 49 

49 25' 02" 
Add Parallax, ... 00 00 06 

True altitude 0's center, . 49 25' 08" 

Ex. 2. With the Artificial Horizon. 

Altitude of 0's upper limb above the image in the artificial ho- 
rizon, 100 2' 47". 

True altitude, 50 01' 23."5 

's semi-diameter, . . . 00 15 50. 

49 45' 33."5 
Refraction, 00 00 48. 

49 44' 45."5 
Parallax, 00 00 05. 

True altitude of 's center, . . . 49 44' 5Q."5 

* Woodhouse's Ast. p. 774. 



132. Given the sun's Right Ascension and Declination, to find 
his Longitude and the Obliquity of the Ecliptic. 

Let PCP' (Fig. 20,) represent the celestial meridian that passes 
through the first of Cancer and Capricorn, (the solstitial colure,) 
PP' the axis of the sphere, EQ the equator, E'C the ecliptic, and 
PSP' the declination circle (Art. 
37,) passing through the sun S ; 
then ARS is a right angle, and in 
the right angled spherical triangle 
ARS, are given the right ascension 
AR (Art. 37,) and the declination 
RS, to find the longitude AS and 
the obliquity SAR. 

As longitude and right ascension 
are measured from A, the first point 

of Aries, in the direction AS of the signs, quite round the globe, 
when, of the four quantities mentioned in the problem, the obliquity 
and the declination are given to find the others, we must know 
whether the sun is north, or whether it is south of the equator, the 
longitude being in the one case AS, and in the other, instead of 
AS', it is 360 AS', that is, the supplement of AS'. We must 
also know on which side of the tropic the sun is, for the sun in 
passing from one of the tropics to the equinox, passes through the 
same degrees of declination as it had gone through in ascending 
from the other equinox to the tropic, although its longitude and 
right ascension go on continually increasing. From the 21st of 
March to the 21st of June, while describing the first quadrant 
from the vernal equinox, the declination is north and increasing ; 
north but decreasing, in the second quadrant, until the 23d of 
September ; south and increasing in the third quadrant, until the 
21st of December ; and finally, in the fourth quadrant, south but 
decreasing until the 21st of March. 

Ex. 1. On the 17th of May, the sun's Right Ascension was 
53 38', and his Declination 19 15' 57": required his Longitude 
and the Obliquity of the Ecliptic. 

Young's Spherical Trigonometry, p. 136. Vince's Complete System, Vol. I. 


Applying Napier's rule* to the right angled triangle, ARS, we 

1. Rad. cos. AS=cos. AR cos. RS. 

2. Rad. sin. AR=tan. RS cot. A. -.cot. A= 

tan. RS 

Hence the computation for AS and A is as follows : 

For the Longitude AS. For the Obliquity A. 

cos.AR 53 38' 00" 9.7730185 
cos.RS 19 15 57 9.9749710 

cos.AS 55 57 43 9.7479895 

sin AR 9.9059247 

tan. RS, ar. com. 0.4565209 

cot. A 23 27' 50" 10.3624456 

Ex. 2. On the 31st of March, 1816, the sun's Declination was 
observed at Greenwich to be 4 13' 3H": required his Right 
Ascension, the obliquity of the ecliptic being 23 27' 51". 

- Ans. 9 47' 59". 

Ex. 3. What was the sun's Longitude on the 28th of Novem- 

* The student *s supposed to be acquainted with Spherical Trigonometry ; but to re- 
fresh his memory, we may insert a remark or two. 

It will be recollected that in Napier's rule for the solution of a right angled spherical 
triangle, by means of the Five Circular Parts, we proceed as follows. 

In a right angled spherical triangle we are to recognize but five parts, viz. the three 
sides and the two oblique angles. If we take any one of these as a middle part, the 
two which lie next to it on each side will be adjacent parts. Thus, (in Fig. 21,) taking 
A for a middle part, b and c will be the adjacent parts ; if we take c for the middle part, 
A and B will be the adjacent parts ; if we fig. 21. 

take B for the middle part, c and a will be 
the adjacent parts ; but if we take a for 
the middle part, then as the angle C is 
not considered as one of the circular parts, 
B and b are the adjacent parts ; and, last- 

ly, if b is the middle part, then the adja- ^ 

cent parts are A and a. The two parts immediately beyond the adjacent parts on each 
side, still disregarding the right angle, are called the opposite parts ; thus if A is the 
middle part, the opposite parts are a and B. Napier's rule is as follows : 

Radius into the sine of the middle part, equals the product of the tangents of the 
adjacent extremes, or of the cosines of the opposite extremes. 

(The corresponding vowels are marked to aid the memory.) This rule is modified 
by using the complements of the two angles and the hypothenuse instead of the parts 
themselves. Thus instead of rad.Xsin. A, we say rad.Xcos. A, when A is the middle 
part ; and rad.Xcos. AB, when the hypothenuse is the middle part. 

Examples. 1. In the right angled triangle ABC, are given the two perpendicular 
sides, viz. a=48 24' 10", 6=59 38' 27", to find the hypothenuse c. The hypothenuse 
being made the middle part, the other sides become the opposite parts, being separated 


her, 1810, when his Declination was 21 16' 4", and his Right 
Ascension, in time, 16h. 14m. 58.4s.? 

Ans. 245 39' 10". 

Ex. 4. The sun's Longitude being 8s. 7 40' 56", and the Ob- 
liquity 23 2V 42i", what was the Right Ascension in time? 

Ans. 16h. 23m. 34s. 

133. Given the surfs Declination to find the time of his Rising 
and Setting at any place whose latitude is known. 

Let PEP' (Fig. 22,) represent the meridian of the place, Z 
being the zenith, and HO the horizon ; and let LL' be the appa- 
rent path of the sun on the proposed 
day, cutting the horizon in S. Then 
the arc EZ will be the latitude of the 
place, and consequently EH, or its 
equal QO, will be the co-latitude, and 
this measures the angle OAQ ; also 
RS will be the sun's declination, and 
AR expressed in time will be the time 
of rising before 6 o'clock. For it is 
evident that it will be sunrise when 

the sun arrives at the horizon at S ; but PP' being an hour circle 
whose plane is perpendicular to the meridian, (and of course pro- 
jected into a straight line on the plane of projection,) the time the 
sun is passing from S to S' taken from the time of describing S'L, 
which is six hours, must be the time from midnight to sunrise. 
But the time of describing SS ; is measured on the corresponding 
arc of the equinoctial AR. 

In the right angled triangle ARS, we have the declination RS,^,\> * 
and the angle A to find AR. Therefore, , i>& (*& : 

Rad. xsin. AR^cot. A xtan. RS. 

from the middle part by the angles A and B. Hence, rad. cos. c=cos. a cos. b .. cos. c= 
cos., cos. 6 

2. In the spherical triangle, right angled at C, are given two perpendicular sides, 
viz. a=116 30' 43", i=29 41' 32", to find the angle A. 

Here, the required angle is adjacent to one of the given parts, viz. 6, which make 
the middle part. Then, 

Rad.xsin, 6=cot A tan. a .-.cot. A= rad - Xsin ' 6 =76 o 7 / U /r. 

tan. a. 



Ex. 1. Required the time of sunrise at latitude 52 13' N 
when the sun's declination is 23 28'. 

Rad 10. 

* 10.1105786 
L 9.6376106 

Cot. A or tan. 52 13' 
Tan. BS= 23 28' 

Sin. 34 03 21i" 



2h. 16' 13" 25'" ; 


3h. 43' 46" 35"'= the time after midnight, and of 
course the time of rising. 

Ex. 2. Required the time of sunrise at latitude 57 2' 54" N. 
when the sun's declination is 23 28' N. 

Ans. 3h. llm. 49s. 

Ex. 3. How long is the sun above the horizon in latitude 58 
12' N. when his declination is 18 40' S. ? 

Ans. 7h. 35m. 52s. 

134. Given the Latitude of the place, and the Declination of a 
heavenly body, to determine its Altitude and Azimuth when on the 
six o'clock hour circle. 

Let HZO (Fig. 23,) be the meridian of the place, Z the zenith. 

Fig. 23. 

HO the horizon, S the place of 
the object on the 6 o'clock hour 
circle PSP', which of course cuts 
the equator in the east and west 
points, and ZSB the vertical cir- 
cle passing through the body. 
Then in the right,angled triangle 
SBA, the given quantities are 
AS, which is the declination, 
and the arc OP or angle SAB, 
the latitude of the place, to find 
the altitude BS, and the azimuth 
BO, or the amplitude AB, which is its complement. 

Ex. 1. What were the altitude and azimuth of Arcturus, when 

Degrees are converted into hours by multiplying by 4 and dividing by 60. 



upon the six o'clock hour circle of Greenwich, lat. 51 28' 40" N. 
on the first of April, 1822 ; its declination being 20 6' 50" N.? 

For the Attitude. 

Rad. sin. BS=sin. AS sin. A 
Rad. . . . 10. 
Sin. 20 06' 50" 9.5364162 
Sin. 51 28 40 9.8934103 

Sin. 15 36 27 


For the Azimuth. 

Rad. cos. A=cot. BO cot. AS 
Cot. 20 06' 50" 10.4362545 
Cos. 51 28 40 9.7943612 

Rad. N '. 10. 

Cot. 77 09' 04' 


Ex. 2. At latitude 62 12' N. the altitude of the sun at 6 o'clock 
in the morning was found to be 18 20' 23": required his declina- 
tion and azimuth. 

Ans. Dec. 20 50' 12" N. Az. 79 56' 4". 

135. The Latitudes and Longitudes of two celestial objects be- 
ing given, to find their Distance apart. 

Let P (Fig. 24,) represent the pole of the ecliptic, and PS, PS', 
two arcs of celestial latitude (Art. 37,) drawn to the two objects 
SS' ; then will these arcs represent the Fig. 24 

co-latitudes, the angle P will be the 
difference of longitude, and the arc SS' 
will be the distance sought. Here we 
have the two sides and the included 
angle given to find the third side. By 
Napier's Rules for the solution of oblique angled spherical triangles, 
(see Spherical Trigonometry,) the sum and difference of the two 
angles opposite the given sides may be found, and thence the an- 
gles themselves. The required side may then be found by the theo- 
rem, that the sines of the sides are as the sines of their opposite 
angles.* The computation is omitted here on account of its great 
length. If P be the pole of the equator instead of the ecliptic, 
then PS and PS' will represent arcs of co-declination, and the 
angle P will, denote difference of right ascension. From these 
data, also, we may therefore derive the distance between any two 
stars. Or, finally, if P be the pole of the horizon, the angle at P 

* More concise formulae for the solution of this case may be found in Young's Tri- 
gonometry, p. 99. Francoeur's Uranography, Art. 330. Dr. Bowditch's Practical 
Navigator, p. 436. 


will denote difference of azimuth, and the sides PS, PS', zenith 
distances, from which the side SS' may likewise be determined. 


136. We have already shown, (Art. 8,) that the figure of the 
earth is nearly globular ; but since the semi-diameter of the earth 
is taken as the base line in determining the parallax of the heav- 
enly bodies, and lies therefore at the foundation of all astronomi- 
cfrl measurements, it is very important that it should be ascertained 
with the greatest possible exactness. Having now learned the 
use of astronomical instruments, and the method of measuring 
arcs on the celestial sphere, we are prepared to understand the 
methods employed to determine the exact figure of the earth. 
This element is indeed, ascertained in four different ways, each 
of which is independent of all the rest, namely, by investigating 
the effects of the centrifugal force arising from the revolution of 
the earth on its axis by measuring arcs of the meridian by 
experiments with the pendulum and by the unequal action of the 
earth on the moon, arising from the redundance of matter about 
the equatorial regions. We will briefly consider each of these 

137. First, the known effects of the centrifugal force, would give 
to the earth a spheroidal figure, elevated in the equatorial, and flat- 
tened in the polar regions. 

Had the earth been originally constituted (as geologists sup- 
pose) of yielding materials, either fluid or semi-fluid, so that 
its particles could obey their mutual attraction, while the body 
remained at rest it would spontaneously assume the figure of a 
perfect sphere ; as soon, however, as it began to revolve on its 
axis, the greater velocity of the equatorial regions would give to 
them a greater centrifugal force, and cause the body to swell out 
into the form of an oblate spheroid.* Even had the solid part of 
the earth consisted of unyielding materials and been created a 
perfect sphere, still the waters that covered it would have receded 
from the polar and have been accumulated in the equatorial re- 

* See a good explanation of this subject in the Edinburgh Encyclopaedia, II. 665. 



gions, leaving bare extensive regions on the one side, and ascend- 
ing to a mountainous elevation on the other. 

On estimating from the known dimensions of the earth and 
the velocity of its rotation, the amount of the centrifugal force in 
different latitudes, and the figure of equilibrium which would 
result, Newton inferred that the earth must have the form of an 
oblate spheroid before the fact had been established by observa- 
tion ; and he assigned nearly the true ratio of the polar and equa- 
torial diameters. 

138. Secondly, the spheroidal figure of the earth is proved, by 
actually measuring the length of a degree on the meridian in differ- 
ent latitudes. 

Were the earth a perfect sphere, the section of it made by a 
plane passing through its center in any direction would be a per- 
fect circle, whose curvature would be equal in all parts ; but if 
we find by actual observation, that the curvature of the section is 
not uniform, we infer a corresponding departure in the earth from 
the figure of a perfect sphere. This task of measuring portions of 
the meridian, has been executed in different countries by means 
of a system of triangles with astonishing accuracy.* The result 
is, that the length of a degree increases as we proceed from the 
equator towards the pole, as may be seen from the following table : 

Places of observation. 


Length of a degree in miles. 


00 00' 00" 
39 12 00 
43 01 00 
46 12 00 
51 29 54| 
66 20 10 


Combining the results of various measurements, the dimensions 
of the terrestrial spheroid are found to be as follows : f 

Equatorial diameter, . . . 7925.308 

Polar diameter, .... 7898.952 

Mean diameter, .... 7912.130 

The difference between the greatest and least, is 26.356=^ 

* See Day's Trigonometry. 

t Bessel. 


of the greatest. This fraction (^I T ) is denominated the elliplicity 
of the earth, being the excess of the transverse over the conjugate 
axis, on the supposition that the section of the earth coinciding 
with the meridian, is an ellipse : and that such is the case, is 
proved by the fact that calculations on this hypothesis, of the 
lengths of arcs of the meridian in different latitudes, agree nearly 
with the lengths obtained by actual measurement. 

139. Thirdly, the figure of the earth is shown to be spheroidal, by 
observations with the pendulum. 

The use of the pendulum in determining the figure of the 
earth, is founded upon the principle that the number of vibra- 
tions performed by the same pendulum, when acted on by differ 
ent forces, varies as the square root of the forces.* Hence, by 
carrying a pendulum to different parts of the earth, and counting 
the number of vibrations it performs in a given time, we obtain 
the relative forces of gravity at those places, and this leads to a 
knowledge of the relative distance of each place from the center 
of the earth, and finally, to the ratio between the equatorial and 
the polar diameters. 

140. Fourthly, that the earth is of a spheroidal figure, is infer- 
red from tJie motions of the moon. 

These are found to be affected by the excess of matter about 
the equatorial regions, producing certain irregularities in the lunar 
motions, the amount of which becomes a measure of the excess 
itself, and hence affords the means of determining the earth's 
ellipticity. This calculation has been made by the most profound 
mathematicians, and the figure deduced from this source corres- 
ponds very nearly to that derived from the several other indepen 
dent methods. 

We thus have the shape of the earth established upon the most 
satisfactory evidence, and are furnished with a starting point from 
which to determine various measurements among the heavenly 
bodies. x 

141. The density of the earth compared with water, that is, its 

* Mechanics, Art. 183. 


specific gravity, is 5.* The density was first estimated by Dr 
Hutton, from observations made by Dr. Maskelyne, Astronomer 
Royal, on Schehallien, a mountain of Scotland, in the year 1774. 
Thus, let M (Fig. 25,) represent Fig. 25. 

the mountain, D, B, two stations 
on opposite sides of the moun- 
tain, and I a star; and let IE 
and IG be the zenith distances as 
determined by the differences of 
latitudes of the two stations. But 
the apparent zenith distances as 
determined by the plumb line 
are IE' and IG'. The deviation 
towards the mountain on each 
side exceeded 7".f The attrac- 
tion of the mountain being ob- 
served on both sides of it, and 
its mass being computed from a number of sections taken in all 
directions, these data, when compared with the known attraction 
and magnitude of the earth, led to a knowledge of its mean den- 
sity. According to Dr. Hutton, this is to that of water as 9 to 2 ; 
but later and more accurate estimates have made the specific 
gravity of the earth as stated above. But this density is nearly 
double the average density of the materials that compose the ex- 
terior crust of the earth, showing a great increase of density 
towards the center. 

The density of the earth is an important element, as we shall 
find that it helps us to a knowledge of the density of each of the 
other members of the solar system. 

Daily, Ast Tables, p. 21. 

t Robison's Phys. Ast. 


142. HAVING considered the Earth, in its astronomical relations, 
and the Doctrine of the Sphere, we proceed now to a survey of 
the Solar System, and shall treat successively of the Sun, Moon, 
Planets, and Comets. 



143. THE figure which the sun presents to us is that of a per- 
fect circle, whereas most of the planets exhibit a disk more or less 
elliptical, indicating that the true shape of the body is an oblate 
spheroid. So great, however, is the distance of the sun, that a 
line 400 miles long would subtend an angle of only 1" at the eye, 
and would therefore be the least space that could be measured. 
Hence, were the difference between two conjugate diameters of 
the sun any quantity less than this, we could not determine by 
actual measurement that it existed at all. Still we learn from 
theoretical considerations, founded upon the known effects of cen- 
trifugal force, arising from the sun's revolution on his axis, that 
his figure is not a perfect sphere, but is slightly spheroidal.* 

144. The distance of the sun from the earth, is nearly 95,000,000 
miles. For, its horizontal parallax being 8."6, (Art. 86,) and the 
semi-diameter of the earth 3956 miles, 

Sin. 8."6 : 3956 : : Rad. : 95,000,000 nearly. In order to form 
some faint conception at least of this vast distance, let us reflect 
that a railway car, moving at the rate of 20 miles per hour, would 
require more than 500 years to reach the sun. 

* See Mecanique Celeste, III, 165. Delambre, t, I, p. 48a 


145. The apparent diameter of the sun may be found either by 
the Sextant, (Art. 129,) by an instrument called the Heliometer, 
specially designed for measuring its angular breadth, or by the time 
it occupies in crossing the meridian. If, for example, it occupied 
4 m , its angular diameter would be 1. It in fact occupies a little 
more than 2 m , and hence its apparent diameter is a little more than 
half a degree, (32' 3"). Having the distance and angular diameter, 
we can easily find its linear diameter. Let E (Fig. 26,) be the 
earth, S the sun, ES a line drawn to the Fig. 26. 

center of the disk, and EC a line drawn 
touching the disk at C. Join SC ; then 

Rad. : ES (95,000,000) : : sin. 16' l."5 : 
442840=semi-diameter, and 885680=diam- \ ^fc 

eter. And - =112 nearly ; that is, it 

would require one hundred and twelve bo- 
dies like the earth, if laid side by side, to 
reach across the diameter of the sun; and a 
ship sailing at the rate of ten knots an hour, 
would require more than ten years to sail 
across the solar disk. Since spheres are to 
each other as the cubes of their diameters, 

I 3 : 112 3 : : 1 : 1,400,000 nearly; that is, the sun is about 
1,400,000 times as large as the earth. The distance of the moon 
from the earth being 237,000 miles, were the center of the sun 
made to coincide with the center of the earth, the sun would ex- 
tend every way from the earth nearly twice as far as the moon. 

146. In density, the sun is only one fourth that of the earth, 
being but a little heavier than water (Art. 141) ; and since the 
quantity of matter, or mass of a body, is proportioned to its mag- 
nitude and density, hence, 1,400,000 x = 350,000, that is, the 
quantity of matter in the sun is three hundred and fifty thousand 
(or, more accurately, 354,936) times as great as in the earth. Now 
the weight of bodies (which is a measure of the force of gravity) 
varies directly as the quantity of matter, and inversely as the 
square of the distance. A body, therefore, would weigh 350,000 
times as much on the surface of the sun as on the earth, if the 

72 THE SUN. 

distance of the center of force were the same in both cases ; but 
since the attraction of a sphere is the same as though all the mat- 
ter were collected in the center, consequently, the weight of a 
body, so far as it depends on its distance from the center of force, 
would be the square of 112 times less at the sun than at the earth. 
Or, putting W for the weight at the earth, and W for the weight 
at the sun, then 

Hence a body would weigh nearly 28 times as much at the sun 
as at the earth. A man weighing 200 Ibs. would, if transported 
to the surface of the sun, weigh 5,580 Ibs., or nearly 2i tons. To 
lift one's limbs, would, in such a case, be beyond the ordinary 
power of the muscles. At the surface of the earth, a body falls 
through IGjL feet in a second ; and since the spaces are as the 
velocities, the times being equal, and the velocities as the forces, 
therefore a body would fall at the sun in one second, through 
16 T V x 27 T 9 o = 448.7 feet. 


147. The surface of the sun, when viewed with a telescope, 
usually exhibits dark spots, which vary much, at different times, 
in number, figure, and extent. One hundred or more, assembled 
in several distinct groups, are sometimes visible at once on the 
solar disk. The solar spots are commonly very small, but 
occasionally a spot of enormous size is seen occupying an extent 
of 50,000 miles or more in diameter. They are sometimes 
even visible to the naked eye, when the sun is viewed through 
colored glass, or when near the horizon, it is seen through light 
clouds or vapors. When it is recollected that 1" of the solar 
disk implies an extent of 400 miles, (Art. 143,) it is evident that a 
space large enough to be seen by the naked eye, must cover a very 
large extent. 

A solar spot usually consists of two parts, the nucleus and the 
umbra, (Fig. 27.) The nucleus is black, of a very irregular shape, 
and is subject to great and sudden changes, both in form and size. 
Spots have sometimes seemed to burst asunder, and to project frag- 
ments in different directions. The umbra is a wide margin of lighter 


shade, and is commonly of greater Fi - 27 - 

extent than the nucleus. The spots 
are usually confined to a zone ex- 
tending across the central regions 
of the sun, riot exceeding 60 in 
breadth. When the spots are ob- 
served from day to day, they are 
seen to move across the disk of the 
sun, occupying about two weeks in 
passing from one limb to the other. 
After an absence of about the same 
period, the spot returns, having taken 27d. 7h. 37m. in the entire 

148. The spots must be nearly or quite in contact with the body 
of the sun. Were they at any considerable distance from it, the 
time during which they would be seen on the solar disk, would 
be less than that occupied in the remainder of the revolution. 
Thus, let S (Fig. 28,) be the sun, E the earth, and abc the path 
of the body, revolving about the sun. 
Unless the spot were nearly or quite 
in contact with the body of the sun, 
being projected upon his disk only 
while passing from b to c, and being 
invisible while describing the arc cab, 
it would of course be out of sight lon- 
ger, than in sight, whereas the two pe- 
riods are found to be equal. Moreover, 
the lines which all the solar spots de- 
scribe on the disk of the sun, are found 
to be parallel to each other, like the 
circles of diurnal revolution around the 
earth ; and hence it is inferred that 
they arise from a similar cause, namely, 
the revolution of the sun on his aocis, 
a fact which is thus made known to 


But although the spots occupy about 27 days in passing from 


74 THE SUN. 

one limb of the sun around to the same limb again, yet this is not 

the period of the sun's revolution on his axis, but exceeds it by 

nearly two days. For, let AA'B (Fig. 29,) represent the sun, and 

EE'M the orbit of the earth. When the earth is at E, the 

visible disk of the sun will be AA'B ; 

and if the earth remained stationary at 

E, the time occupied by a spot after 

leaving A until it returned to A, would 

be just equal to the time of the sun's 

revolution on his axis. But during the 

27| days in which the spot has been 

performing its apparent revolution, the 

earth has been advancing in his orbit 

from E to E', where the visble disk of 

the sun is A'B'. Consequently, before 

the spot can appear again on the limb from which it set out, it 

must describe so much more than an entire revolution as equals 

the arc AA', which equals the arc EE'. Hence, 

365d. 5h. 48m.+27d. 7h. 37m. : 365d. 5h. 48m. : : 27d. 7h. 37m. : 
25d. 9h. 59m.=the time of the sun's revolution on his axis. 

149. If the path which the spots appear to describe by the re- 
volution of the sun on his axis left each a visible trace on his sur- 
face, they would form, like the circles of diurnal revolution on the 
earth, so many parallel rings, of which that which passed through 
the center would constitute the solar equator, while those on each 
side of this great circle would be small circles, corresponding to 
parallels of latitude on the earth. Let us conceive of an artifi- 
cial sphere to represent the sun, having such rings plainly marked 
on its surface. Let this sphere be placed at some distance from 
the eye, with its axis perpendicular to the axis of vision, in which 
case the equator would coincide with the line of vision, and its 
edge be presented to the eye. It would therefore be projected in- 
to a straight line. The same would be the case with all the small- 
er rings, the distance being supposed such that the rays of light 
come from them all to the eye nearly parallel. Now let the axis, 
instead of being perpendicular to the line of vision, be inclined to 
that line, then all the rings being seen obliquely would be projected 


into ellipses. If, however, while the sphere remained in a fixed 
position, the eye were carried around it, (being always in the same 
plane,) twice during the circuit it would be in the plane of the 
equator, and project this and all the smaller circles into straight 
lines ; and twice, at points 90 distant from the foregoing posi- 
tions, the eye would be at a distance from the planes of the rings 
equal to the inclination of the equator of the sphere to the line of 
vision. Here it would project the rings into wider ellipses than 
at other points ; and the ellipses would become more and more 
acute as the eye departed from either of these points, until they 
vanished again into straight lines. 

150. It is in a similar manner that the eye views the paths de- 
scribed by the spots on the sun. If the sun revolved on an axis 
perpendicular to the plane of the earth's orbit, the eye being situ- 
ated in the plane of revolution, and at such a distance from the 
sun that the light comes to the eye from all parts of the solar 
disk nearly parallel, the paths described by the spots would be 
projected into straight lines, and each would describe a straight 
line across the solar disk, parallel to the plane of revolution. But 
the axis of the sun is inclined to the ecliptic about 7 from a per- 
pendicular, so that usually all the circles described by the spots are 
projected into ellipses. The breadth of these, however, will vary 
as the eye, in the annual revolution, is carried around the sun, and 
when the eye comes into the plane of the rings, as it does twice a 
year, they are projected into straight lines, and for a short time a 
spot seems moving in a straight line inclined to the plane of the 
ecliptic 7. The two points where the sun's equator cuts the 
ecliptic are called the sun's nodes. The longitudes of the nodes 
are 80 7' and 260 7', and the earth passes through them about 
the 12th of December, and the llth of June. It is at these times 
that the spots appear to describe straight lines. We have men- 
tioned the various changes in the apparent paths of the solar spots, 
which arise from the inclination of the sun's axis to the plane of 
the ecliptic ; but it was in fact by first observing these changes, 
and proceeding in the reverse order from that which we have pur- 
sued, that astronomers ascertained that the sun revolves on his 
axis, and that this axis is inclined to the ecliptic 82f. 



151. With regard to the cause of the solar spots, various hypo- 
theses have been proposed, none of which is entirely satisfactory. 
That which ascribes their origin to volcanic action, appears to us 
the most reasonable.* 

Besides the dark spots on the sun, there are also seen, in dif- 
ferent parts, places that are brighter than the neighboring por- 
tions of the disk. These are called faculce. Other inequalities 
are observable in powerful telescopes, all indicating that the sur- 
face of the sun is in a state of constant and powerful agitation. 


152. The Zodiacal Light is a faint light resembling the tail of 
a comet, and is seen at certain seasons of the year following the 
course of the sun after evening twilight, or preceding his approach 
in the morning sky. Figure 30 represents its appearance as seen 
in the evening in March, 1836. The following are the leading 
facts respecting it. 

1 . Its form is that of a luminous Fig. 30. 
pyramid, having its base towards 

the sun. It reaches to an immense 
distance from the sun, sometimes 
even beyond the orbit of the earth. 
It is brighter in the parts nearer the 
sun than in those that are more 
remote, and terminates in an ob- 
tuse apex, its light fading away by 
insensible gradations, until it be- 
comes too feeble for distinct vision. 
Hence its limits are, at the same 
time, fixed at different distances 
from the sun by different observers, 
according to their respective powers 
of vision. 

2. Its aspects vary very much with the different seasons of the 
year. About the first of October, in our climate, (Lat. 41 18',) 

* In the system of instruction in Yale College, subjects of this kind are discussed 
in a course of astronomical lectures, addressed to the class after they have finished the 
perusal of the text-book. 


it becomes visible before the dawn of day, rising along north of 
the ecliptic, and terminating above the nebula of Cancer. About 
the middle of November, its vertex is in the constellation Leo. 
At this time no traces of it are seen in the west after sunset, but 
about the first of December it becomes faintly visible in the west, 
crossing the Milky Way near the horizon, and reaching from the 
sun to the head of Capricornus, forming, as its brightness increases, 
a counterpart to the Milky Way, between which on the right, 
and the Zodiacal Light on the left, lies a triangular space embra- 
cing the Dolphin. Through the month of December, the Zodi- 
acal Light is seen on both sides of the sun, namely, before the 
morning and after the evening twilight, sometimes extending 50 
westward, and 70 eastward of the sun at the same time. After 
it begins to appear in the western sky, it increases rapidly from 
night to night, both in length and brightness, and withdraws itself 
from the morning sky, where it is scarcely seen after the month 
of December, until the next October. 

3. The Zodiacal Light moves through the heavens in the order of 
the signs. It moves with unequal velocity, being sometimes sta- 
tionary and sometimes retrograde, while at other times it ad- 
vances much faster than the sun. In February and March, it is 
very conspicuous in the west, reaching to the Pleiades and be- 
yond ; but in April it becomes more faint, and nearly or quite dis- 
appears during the month of May. It is scarcely seen in this lat- 
itude during the summer months. 

4. It is remarkably conspicuous at certain periods of a few 
years, and then far a long interval almost disappears. 

5. The Zodiacal Light was formerly field to be the atmosphere of 
the sun.* But La Place has shown that the solar atmosphere 
could never reach so far from the sun as this light is seen to ex- 
tend.f It has been supposed by others to be a nebulous body 
revolving around the sun. The idea has been suggested, that the 
extraordinary Meteoric Showers, which at different periods visit 
the earth, especially in the month of November, may be derived 
from this body.J 

* Mairan, Memoirs French Academy, for 1733. t Mec. Celeste, III, 525. 

t See note on " Meteoric Showers," at the end of the volume. 




153. THE revolution of the earth around the sun once a year, 
produces an apparent motion of the sun around the earth in the 
same period. When bodies are at such a distance from each 
other as the earth and the sun, a spectator on either would pro- 
ject the other body upon the concave sphere of the heavens, al- 
ways seeing it on the opposite side of a great circle, 180 from 
himself. Thus when the earth arrives at Libra (Fig. 11,) we see 
the sun in the opposite sign Aries. When the earth moves from 
Libra to Scorpio, as we are unconscious of our own motion, the 
sun it is that appears to move from Aries to Taurus, being always 
seen in the heavens, where a line drawn from the eye of the spec- 
tator through the body meets the concave sphere of the heavens. 
Hence the line of projection carries the sun forward on one side 
of the ecliptic, at the same rate as the earth moves on the oppo- 
site side ; and therefore, although we are unconscious of our own 
motion, we can read it from day to day in the motions of the sun. 
If we could see the stars at the same time with the sun, we could 
actually observe from day to day the sun's progress through them, 
as we observe the progress of the moon at night ; only the sun's 
rate of motion would be nearly fourteen times slower than that 
of the moon. Although we do not see the stars when the sun is 
present, yet after the sun is set, we can observe that it makes daily 
progress eastward, as is apparent from the constellations of the 
Zodiac occupying, successively, the western sky after sunset, 
proving that either all the stars have a common motion westward 
independent of their diurnal motion, or that the sun has a motion 
past them, from west to east. We shall see hereafter abundant 
evidence to prove, that this change in the relative position of the 
sun and stars, is owing to a change in the apparent place of the 
sun, and not to any change in the stars. 


154. Although the apparent revolution of the sun is in a direc- 
tion opposite to the real motion of the earth, as regards absolute 
space, yet both are nevertheless from west to east, since these 
terms do not refer to any directions in absolute space, but to the 
order in which certain constellations (the constellations of the 
Zodiac) succeed one another. The earth itself, on opposite sides 
of its orbit, does in fact move towards directly opposite points of 
space ; but it is all the while pursuing its course in the order of 
the signs. In the same manner, although the earth turns on its 
axis from west to east, yet any place on the surface of the earth 
is moving in a direction in space exactly opposite to its direction 
twelve hours before. If the sun left a visible trace on the face 
of the sky, the ecliptic would of course be distinctly marked on 
the celestial sphere as it is on an artificial globe ; and were the 
equator delineated in a similar manner, (by any method like that 
supposed in Art. 46,) we should then see at a glance the relative 
position of these two circles, the points where they intersect one 
another constituting the equinoxes, the points where they are at 
the greatest distance asunder, or the solstices, and various other 
particulars, which, for want of such visible traces, we are now 
obliged to search for by indirect and circuitous methods. It will 
even aid the learner to have constantly before his mental vision, 
an imaginary delineation of these two important circles on the 
face of the sky. 

155. The method of ascertaining the nature and position of the 
earth's orbit, is by observations on the sun's Decimation and Right 

The exact declination of the sun at any time is determined 
from his meridian altitude or zenith distance, the latitude of the 
place of observation being known, (Art. 37.) The instant the 
center of the sun is on the meridian, (which instant is given by 
the transit instrument,) we take the distance of his upper and 
that of his lower limb from the zenith : half the sum of the two 
observations corrected for refraction, gives the zenith distance of 
the center. This result is diminished for parallax, (Art. 84,) and 
we obtain the zenith distance as it would be if seen from the 
center of the earth. The zenith distance being known, the de- 

80 THE SUN. 

ciination is readily found, by subtracting that distance from the 
latitude. By thus taking the sun's declination for every day of 
the year at noon, and comparing the results, we learn its motion 
to and from the equator. 

156. To obtain the motion in right ascension, we observe, with 
a transit instrument, the instant when the center of the sun is on 
the meridian. Our sidereal clock gives us the right ascension in 
time (Art. 124,) which we may easily, if we choose, convert into 
degrees and minutes, although it is more common to express right 
ascension by hours, minutes, and seconds. The differences of 
right ascension from day to day throughout the year, give us the 
sun's annual motion parallel to the equator. From the daily re- 
cords of these two motions, at right angles to each other, arran- 
ged in a table,* it is easy to trace out the path of the sun on the 
artificial globe ; or to calculate it with the greatest precision by 
means of spherical triangles, since the declination and right ascen- 
sion constitute two sides of a right angled spherical triangle, the 
corresponding arc of the ecliptic, that is, the longitude, being the 
third side, (Art. 132.) By inspecting a table of observations, 
we shall find that the declination attains its greatest value on 
the 22d of December, when it is 23 27' 54" south ; that from 
this period it diminishes daily and becomes nothing on the 21st 
of March ; that it then increases towards the north, and reaches 
a similar maximum at the northern tropic about the 22d of June ; 
and, finally, that it returns again to the southern tropic by gra- 
dations similar to those which marked its northward progress. A 
table of observations also would show us, that the daily differences 
of declination are very unequal ; that, for several days, when the 
sun is near either tropic, its declination scarcely varies at all ; 
while near the equator, the variations from day to day are very 
rapid, a fact which is easily understood, when we reflect, that 
at the solstices the equator and the ecliptic are parallel to each 
other,f both being at right angles to the meridian ; while at the 

* Such a table may be found in Blot's Astronomy, in Delambre, and in most collec- 
tions of Astronomical Tables. 

t Or, more properly, the tangents of the two circles (which denote the directions of 
the curves at those points) are parallel. 


equinoxes, the ecliptic departs most rapidly from the direction of 
the equator. 

On examining, in like manner, a table of observations of the 
right ascension, we find that the daily differences of right ascen- 
sion are likewise unequal ; that the mean of them all is 3 m 56 s , 
or 236 s , but that they have varied between 215 s and 266 s . On 
examining, moreover, the right ascension at each of the equi- 
noxes, we find that the two records differ by 180; which proves 
that the path of the sun is a great circle, since no other would 
bisect the equinoctial as this does. 

157. The obliquity of the ecliptic is equal to the sun's greatest 
declination. For, by article 22, the inclination of any two great 
circles is equal to their greatest distance asunder, as measured on 
the sphere. The obliquity of the ecliptic may be determined 
from the sun's meridian altitude, or zenith distance, on the clay 
of the solstice. The exact instant of the solstice, however, will 
not of course occur when the sun is on the meridian, but may 
happen at some other meridian ; still, the changes of declination 
near the solstice are so exceedingly small, that but a slight error 
can result from this source. The obliquity may also be found, 
without knowing the latitude, by observing the greatest and least 
meridian altitudes of the sun, and taking half the difference. 
This is the method practiced in ancient times by Hipparchus. 
(Art. 2.) On comparing observations made at different periods 
for more than two thousand years, it is found, that the obliquity 
of the ecliptic is not constant, but that it undergoes a slight dimi- 
nution from age to age, amounting to 52" in a century, or about 
half a second annually. We might apprehend that by successive 
approaches to each other the equator and ecliptic would finally 
coincide ; but astronomers have ascertained by an investigation, 
founded on the principles of universal gravitation, that this varia- 
tion is confined within certain narrow limits, and that the obli- 
quity, after diminishing for some thousands of years, will then 
increase for a similar period, and will thus vibrate for ever about 
a mean value. 

158. The dimensions of the earth 9 s orbit, when compared with its 
own magnitude, are immense. 


85? THE SUN. 

Since the distance of the earth from the sun is 95,000,000 
miles, and the length of the entire orbit nearly 600,000,000 miles, 
it will be found, on calculation, that the earth moves 1,640,000 
miles per day, 68,000 miles per hour, 1,100 miles per minute, and 
nearly 19 miles every second, a velocity nearly fifty times as great 
as the maximum velocity of a cannon ball. A place on the earth's 
equator turns, in the diurnal revolution, at the rate of about 1,000 
miles an hour and T \ of a mile per second. The motion around 
the sun, therefore, is nearly 70 times as swift as the greatest mo- 
tion around the axis. 


159. The change of seasons depends on two causes, (1) the ob- 
liquity of the ecliptic, and (2) the earth's axis always remaining 
parallel to itself. Had the earth's axis been perpendicular to the 
plane of its orbit, the equator would have coincided with the 
ecliptic, and the sun would have constantly appeared in the equa- 
tor. To the inhabitants of the equatorial regions, the sun would 
always have appeared to move in the prime vertical ; and to the 
inhabitants of either pole, he would always have been in the ho- 
rizon. But the axis being turned out of a perpendicular direc- 
tion 23 28', the equator is turned the same distance out of the 
ecliptic ; and since the equator and ecliptic are two great circles 
which cut each other in two opposite points, the sun, while per- 
forming his circuit in the ecliptic, must evidently be once a year 
in each of those points, and must depart from the equator of the 
heavens to a distance on either side equal to the inclination of the 
two circles, that is, 23 28'. (Art. 22.) 

160. The earth being a globe, the sun constantly enlightens 
the half next to him,* while the other half is in darkness. The 
boundary between the enlightened and the unenlightened part, is 
called the circle of illumination. When the earth is at one of 
the equinoxes, the sun is at the other, and the circle of illumina- 

* In fact, the sun enlightens a little more than half the earth, since on account of 
his vast magnitude the tangents drawn from opposite sides of the sun to opposite sides 
of the earth, converge to a point behind the earth, as will be seen by and by in the 
representation of eclipses. The amount of illumination also is increased by refraction. 


tion passes through both the poles. When the earth reaches one 
of the tropics, the sun being at the other, the circle of illumina- 
tion cuts the earth so as to pass 23 28' beyond the nearer, and 
the same distance short of the remoter pole. These results would 
not be uniform, were not the earth's axis always to remain parallel 
to itself. The following figure will illustrate the foregoing state- 

Fig. 31. 

Let ABCD represent the earth's place in different parts of its 
orbit, having the sun in the center. Let A, C, be the position of 
the earth at the equinoxes, and B, D, its positions at the tropics, 
the axis ns being always parallel to itself.* At A and C the sun 
shines on both n and s ; and now let the globe be turned round 
on its axis, and the learner will easily conceive that the sun will 
appear to describe the equator, which being bisected by the hori- 

* The learner will remark that the hemisphere towards n is above, and that towards 
* is below the plane of the paper. It is important to form a just conception of the 
position of the axis with respect to the plane of its orbit. 

84 THE SUN. 

zon of every place, of course the day and night will be equal in all 
parts of the globe.* Again, at B when the earth is at the south- 
ern tropic, the sun shines 23 1 beyond the north pole n, and falls 
the same distance short of the south pole s. The case is exactly 
reversed when the earth is at the northern tropic and the sun at 
the southern. While the earth is at one of the tropics, at B for 
example, let us conceive of it as turning on its axis, and we shall 
readily see that all that part of the earth which lies within the 
north polar circle will enjoy continual day, while that within the 
south polar circle will have continual night, and that all other 
places will have their days longer as they are nearer to the en- 
lightened pole, and shorter as they are nearer to the unenlightened 
pole. This figure likewise shows the successive positions of the 
earth at different periods of the year, with respect to the signs, 
and what months correspond to particular signs. Thus the earth 
enters Libra and the sun Aries on the 21st of March, and on the 
21st of June the earth is just entering Capricorn and the sun Can- 

161. Had. the axis of the earth been perpendicular to the plane 
of the ecliptic, then the sun would always have appeared to move 
in the equator, the days would every where have been equal to the 
nights, and there could have been no change of seasons. On the 
other hand, had the inclination of the ecliptic to the equator been 
much greater than it is, the vicissitudes of the seasons would have 
been proportionally greater than at present. Suppose, for instance, 
the equator had been at right angles to the ecliptic, in which case, 
the poles of the earth would have been situated in the ecliptic 
itself; then in different parts of the earth the appearances would 
have been as follows. To a spectator on the equator, the sun as 
he left the vernal equinox would every day perform his diurnal 
revolution in a smaller and smaller circle, until he reached the 
north pole, when he would halt for a moment and then wheel 
about and return to the equator in the reverse order. The pro- 
gress of the sun through the southern signs, to the south pole, 
would be similar to that already described. Such would be the 

* At the pole, the solar disk, at the time of tne equinox, appears bisected by the ho 


appearances to an inhabitant of the equatorial regions. To a 
spectator living in an oblique sphere, in our own latitude for ex- 
ample, the sun while north of the equator would advance continu- 
ally northward, making his diurnal circuits in parallels further and 
further distant from the equator, until he reached the circle of per- 
petual apparition, after which he would climb by a spiral course 
to the north star, and then as rapidly return to the equator. By a 
similar progress southward, the sun would at length pass the circle 
of perpetual occultation, and for some time (which would be 
longer or shorter according to the latitude of the place of obser- 
vation) there would be continual night. 

The great vicissitudes of heat and cold which would attend 
such a motion of the sun, would be wholly incompatible with the 
existence of either the animal or the vegetable kingdoms, and all 
terrestrial nature would be doomed to perpetual sterility and deso- 
lation. The happy provision which the Creator has made against 
such extreme vicissitudes, by confining the changes of the seasons 
within such narrow bounds, conspires with many other express 
arrangements in the economy of nature to secure the safety and 
comfort of the human race. 


162. Thus far we have taken the earth's orbit as a great circle, 
such being the projection of it on the celestial sphere ; but we now 
proceed to investigate its actual figure. 

Were the earth's path a circle, having the sun in the center, the 
sun would always appear to be at the same distance from us ; that 
is, the radius of its orbit, or radius vector, the name given to a line 
drawn from the center of the sun to the orbit of any planet, 
would always be of the same length. But the earth's distance 
from the sun is constantly varying, which shows that its orbit is 
not a circle. We learn the true figure of the orbit, by ascertain- 
ing the relative distances of the earth from the sun at various pe- 
riods of the year. These all being laid down in a diagram, accord- 
ing to their respective lengths, the extremities, on being connected, 
give us our first idea of the shape of the orbit, which appears of 
an oval form, and at least resembles an ellipse ; and, on further 



trial, we find that it has the properties of an ellipse. Thus, let E 
(Fig. 32,) be the place of the earth, and a, b t c, &c. successive po- 
sitions of the sun ; the relative lengths of the lines E#, Eft, &c. be- 
ing known on connecting the points, , b, c, &c. the resulting 
figure indicates the true shape of the earth's orbit. 

Fig. 32. 

163. These relative distances are found in two different ways ; 
first, by changes in the surfs apparent diameter, and, secondly, by 
variations in his angular velocity. Were the variations in the 
sun's horizontal parallax considerable, as is the case with the 
moon's, this might be made the measure of the relative distances, 
for the parallax varies inversely as the distance, (Art. 82) ; but the 
whole horizontal parallax of the sun is only 9", and its variations 
are too slight and delicate, and too difficult to be found, to serve 
as a criterion of the changes in the sun's distance from the earth. 
But the changes in the surfs apparent diameter, are much more 
sensible, and furnish a better method of measuring the relative 
distances of the earth from the sun. By a principle in optics, the 
apparent diameter of an object, at different distances from the 
spectator, is inversely as the distance.* Hence, the apparent 
diameters of the sun, taken at different periods of the year, be- 
come measures of the different lengths of the radius vector. 

* More exactly, the tangent of the apparent diameter is inversely as the distance ; 
but in small angles like those concerned in the present inquiry, the angle itself may be 
taken for the tangent. 


164. The point where the earth, or any planet, in its revolution, 
is nearest the sun, is called its perihelion : the point where it is 
furthest from the sun, its aphelion. The place of the earth's peri- 
helion is known, since there the apparent magnitude of the sun is 
greatest ; and when the sun's magnitude is least, the earth is 
known to be at its aphelion. The sun's apparent diameter when 
greatest is 32' 35."6 ; and when least, 31' 31"; hence the radius 
vector at the aphelion : rad. vector at the perihelion : : 32.5933 : 
31.5167 :: 1.034 : 1. Half of the difference of the two is equal 
to the distance of the focus of the ellipse from the center, a quan- 
tity which is always taken as the measure of the eccentricity of a 
planetary orbit. 

165. The differences of angular velocity in the sun in the dif- 
ferent parts of his apparent revolution, are still more remarkable. 
At the perihelion, the sun moves in twenty-four hours over an arc 
of 61', while at the aphelion he describes in the same time an arc 
of only 57', these being the daily increments of longitude in those 
two points respectively. If the apparent motions of the sun de- 
pended alone on our different distances from him, the angular ve- 
locity would vary inversely as the distance, and the ratio expressed 
by these two numbers would be the same as that of the two num- 
bers which denote the differences of apparent diameter in these 

fil 09 r:qqq 

two points. That is, 2f (=1.07) would equal - ( = 1.034) ; 
) ol.51oT 

but the first fraction is equal to the square of the second, for 1.07= 
1 .034 2 . Hence, the surfs angular velocities are to each other inversely 
as the squares of the distances at the perihelion and the aphelion ; and 
by a similar method, the same is found to be true in all points of 
the revolution. 

The angular velocities, therefore, which can be measured very 
accurately by the daily differences of right ascension and declina- 
tion (Art. 132,) converted into corresponding longitudes, enable 
us to determine the different distances of the earth from the sun 
at various points in the orbit. 

166. Since the arcs described by the earth in any small times, 
as in single days, are inversely as the squares of the distances, con- 



sequently, the distances are inversely as the square roots of the arcs. 
Upon this principle, the relative distances of the earth from the 
sun, in every point of its revolution, may be easily calculated. 
Thus, we have seen that the arcs described by the sun in one day 
at the perihelion and aphelion are as 61 to 57. Hence the distances 
of the earth from the sun at those two points are as -s/57 to \/61, 
or as 1 to 1.034. From twenty-four observations made with the 
greatest care by Dr. Maskelyne at the Royal Observatory of 
Greenwich, the following distances of the earth from the sun are 
determined for each month in the year. 

Time of Observation. Distances. 

January 12-13, 0.98448 

February 17-18, 0.98950 

March 14-15, 0.99622 

April 28-29, 1.00800 

May 15-16, 1.01234 

June 17-18, 1.01654 

Time of Observation. Distances. 

July 18-19, 1.01658 

August 26-27, 1.01042 

September 22-23, 1.00283 

October 24-25, 0.99303 

November 18-20, 0.98746 

December 17-18, 0.98415 

Fig. 33. 

167. The angular velocity being 
inveipely as the square of the distance 
in all parts of the solar orbit, it follows 
that the product of the angle described 
in any given time, by the square of the 
distance, is always the same constant 
quantity. For if of two factors, A x 
B, A is increased as B is diminished, 
the product of A and B is always the 
same. If, therefore, from the sun S 
(Fig. 33,) two radii be drawn to T, 

B, the extremities of the arc described in one day, then ST 2 xTB 
gives the same product in all parts of the orbit.* 

168. The radius vector of the solar orbit describes equal spaces 
in equal times, and in unequal times, spaces proportional to the times. 

Let TB (Fig. 33,) be the arc described by the sun in one day ; 
then, Sector TSB=SB xTB. 

* TB, as seen from the earth, would be projected into a circular arc, equal to the 
measure of the angle at S. 


Taking Sb as any radius, describe the circular arc ab, which is 
the measure of the angle at S. Now, 

Sb : ab : : SB : BT=SBx^ ; and substituting this value of BT 

in the above equation, we have TSB=SBxSBx =iSB 2 x. 

IS0 ho 

But Sb is constant, and the product of SB 2 xo is likewise constant ; 
therefore the sector is always equal to a constant quantity, and 
therefore the radius vector passes over equal spaces in equal 

The sun's orbit may be accurately represented by taking some 
point as the perihelion, drawing the radius vector to that point, 
and, considering this line as unity, drawing other radii making 
angles with each other such that the included areas shall be pro- 
portional to the times, and of a length required by the distance of 
each point as given in the table (Art. 166.) On connecting these 
radii, we shall thus see at once how little the earth's orbit departs 
from a perfect circle. Small as the difference appears between 
the greatest and least distances, yet it amounts to nearly -fa of the 
perihelion distance, a quantity no less than 3,000,000 of miles. 

169. The foregoing method of determining the figure of the 
earth's orbit is founded on observation ; but this figure is subject 
to numerous irregularities, the nature of which cannot be clearly 
understood without a knowledge of the leading principles of Uni- 
versal Gravitation. An acquaintance with these will also be in- 
dispensable to our understanding the causes of the numerous ir- 
regularities, which (as will hereafter appear) attend the motions 
of the moon and planets. To the laws of universal gravitation. 
therefore, let us next apply our attention. 

* Francoeur, Uran., p. 62. 





170. UNIVERSAL GRAVITATION, is that influence by which every 
body in the universe, whether great or small, tends towards every 
other, with a force which is directly as the quantity of matter, and 
inversely as the square of the distance. 

As this force acts as though bodies were drawn towards each 
other by a mutual attraction, the force is denominated attraction ; 
but it must be borne in mind, that this term is figurative, and im- 
plies nothing respecting the nature of the force. 

The existence of such a force in nature was distinctly asserted 
by several astronomers previous to the time of Sir Isaac Newton, 
but its laws were first promulgated by this wonderful man in his 
Principia, in the year 1687. It is related, that while sitting in a 
garden, and musing on the cause of the falling of an apple, he 
reasoned thus :* that, since bodies far removed from the earth fall 
towards it, as from the tops of towers, and the highest mountains, 
why may not the same influence extend even to the moon ; and 
if so, may not this be the reason why the moon is made to revolve 
around the earth, as would be the case with a cannon ball were 
it projected horizontally near the earth with a certain velocity. 
According to the first law of motion, the moon, if not continually 
drawn or impelled towards the earth by some force, would not 
revolve around it, but would proceed on in a straight line. But 
going around the earth as she does, in an orbit that is nearly cir- 
cular, she must be urged towards the earth by some force, which, 
in a given time, may be represented by the versed sine of the arc 
described in that time. For let the earth (Fig. 34,) be at E, and 
let the arc described by the moon in one second of time be Ab. 
Were the moon influenced by no extraneous force, to turn her 
aside, she would have described, not the arc Ab, but the straight 
line AB, and would have been found at the end of the given time 

* Pemberton's View of Newton's Philosophy. 



at B instead of b. She therefore departs from the line in which 
she tends naturally to move, by the line B6, which in small angles 
may be taken as equal to the versed sine Aa. This deviation 
from the tangent must be owing to 
some extraneous force. Does this force 
correspond to what the force of gravi- 
ty exerted by the earth, would be at the 
distance of the moon? Now we know the 
distance of the moon from the earth, and 
of course the circumference of her orbit. 
We also know the time of her revolu- 
tion around the earth. Hence we may 
estimate the length of the arc Ab de- 
scribed in one second ; and knowing 
the arc, we can calculate its versed sine. 
For the moon being 60 times as far from the center of the earth, 
as the surface of the earth is from the center, consequently, since 
the force of gravity decreases as the square of the distance in- 
creases,* the space through which the moon would fall by the 

16yV - .05 inches. 

force of the earth's attraction alone, would be 

60 2 

On calculating the value of the versed sine of the arc described in 
one second, it proves to be the same. Hence gravity, and no other 
force than gravity, causes the moon to circulate around the earth. 

171. By this process it was discovered that the law of gravita- 
tion extends to the moon. By subsequent inquiries it was found 
to extend in like manner to all the planets, and to every member 
of the solar system ; and, finally, recent investigations have shown 
that it extends to the fixed stars. The law of gravitation, there- 
fore, is now established as the grand principle which governs all 
the motions of the heavenly bodies. Hence, nothing can be more 
deserving ^ the attention of the student, than the development of 
the results of this universal law. A few of them only are all that 
can be exhibited in a work like the present : their full develop- 

* Natural Philosophy, Art. 7. That gravity follows the ratio of the inverse square 
of the distance was, however, inferred by Newton from one of Kepler's Laws, to be 
mentioned hereafter. 



ment must be sought for in such great works as the Mecanique 
Celeste of La Place. 

172. If a body revolves about an immovable center of force, and 
is constantly attracted to it, it will always move in the same plane, 
and describe areas about the center proportional to the times.* 

Let S (Fig. 35,) be the center of force, and suppose a body to 
be projected at P in the direction of PQR, and take PQ=QR ; 
then, by the first law of motion, the body would move uniformly 
in the direction PQR, and describe PQ, QR, in the same time, if 
no other force acted upon it. But when the body comes to Q 

Fig. 35. 

let a single impulse act at S, sufficient to draw the body through 
QV, in the time it would have described QR ; and complete the 
parallelogram VQRC, and the body in the same time will describe 
QC ; therefore, PQ, QC, are described in the same time. But 
the triangle SCQ=SRQ=SPQ ; that is, equal areas are described 
in equal times. For the same reason, if a single impulse act at 
C, D, E, &c. at equal intervals of time, the several areas SPQ, 
SQC, SCD, SDE, &c. will all be equal to each other. Now this 

* The learner will remark that what has been before proved (Art. 168,) respecting 
the radius vector of the earth, is here shown to hold good with respect to every body 
which revolves around a center of force ; and the same is true of several other propo- 
sitions demonstrated in this chapter. 



demonstration is independent of any particular dimensions in the 
several triangles, and consequently holds good when they are 
taken indefinitely small, in which case we may consider the force 
as acting, not by separate impulses, but constantly, causing the 
body to describe a curve around S. And as no force acts out of 
the plane SPQ, the whole curve must lie in that plane ; that is, 
the body moves always in the same plane. 

173. If a body describes a curve around a center towards which it 
tends by any force, the angular velocity of the body around that center 
is reciprocally as the square of the distance from it.* 

Let ABE (Fig. 36,) be any curve de- 
scribed about the center S ; draw SA, SB, 
to any two points of the curve A and B ; 
and let AD, BE, be described in indefi- 
nitely small equal times. Join SD and 
SE, and with the center S and distance 
SD, describe a circle meeting SA, SB, SE, 
in F, G, H ; and with the center S and 
distance SE describe a circle meeting SB 

Because AD and BE are described in 
equal times, the triangles ASD, BSE, are 
equal. Hence, (Euc. 15. 6.) 
DF : EK :: BS : ASf :: BS 2 : BSxAS (1) 
SH : SE 
.-. DF 

Fig. 36. 



BS 2 

:AS 2 

SA 2 :BSxAS (2) 




: BS 2 : AS 2 . 

But DF and GH measure the respective angular velocities at 
A and B, while AS and BS represent the distance at the same 
points. Therefore the angular velocities are reciprocally as the 
squares of the distances. J 

174. In the same curve, the velocity, at any point of the curve, 

* It will be remarked that this is a general proposition, of which article 165 affords 
A particular example. 

t DF and EK are considered as the altitudes of the triangles respectively, 
f Stewart's Phys. and Math. Essays. 


varies inversely as the perpendicular drawn from the center of 
force to the tangent at that point. 

Draw SY (Fig. 35,) perpendicular to QP produced ; then the 
area SPQ=iPQ x SY, which varies as PQ x SY /. PQ a 

in the curve described from P, with a constant force, SY becomes 
a perpendicular to the tangent to the curve. But by article 
172, the area described in a given time is constant. Therefore 

SPQ is constant, and V a - ; that is, the velocity varies inverse- 


ly as the perpendicular upon the tangent. Hence, the velocity of 
a revolving body increases as it approaches the center of force. 

175. If equal areas be described about a center in equal times. 
the force must tend towards that center. 

Let SPQ (Fig. 35,)=SQC ; now SPQ-SQR /. SQC-SQR.-. 
CR is parallel to QS. Complete the parallelogram QRCV, and 
by the supposition the body describes QC, in consequence of the 
impulse at Q, and it would have described QR if no such impulse 
had acted ; therefore QV must represent that motion impressed 
at Q, which, in conjunction with the motion QR, can make a body 
describe QC, and QV is directed to S. 

176. Now it appears from article 168, that it is a fact, derived 
from observation, that the earth's radius vector describes equal 
areas in equal times ; and by similar observations the same is 
found to be true of each of the primary planets about the sun, 
and of each of the satellites about its primary. Hence, it is in- 
ferred, that the primary planets all gravitate towards the sun, and 
that the secondary planets all gravitate towards their respective 

It has further been established by observation, (Art. 162,) that 
the planetary orbits are ellipses ; and hence the application of the 
principles of gravitation, so far as respects the sun and planets, 
may be confined to the consideration of the motion of a body in 
an elliptical orbit. 

177. The distance of any planet from the sun at any point in its 



orbit, is to its distance from the superior focus, as the square of its 
velocity at its mean distance from the sun, is to the square of its ve- 
locity at the given point. 

Let ADBE (Fig. 37,) be the orbit of a planet, S the focus in 
which the sun is placed, AB the transverse and DE the conjugate 
axis, C the center, and F the superior focus. Let the planet be 
any where at P ; and draw a tangent to the orbit at P, on which 
from the foci let fall the perpendiculars SG, FH. Draw also DK 
touching the orbit in D, and let SK be perpendicular to it. Let 

Fig. 37. 


the velocity of the planet when at the mean distance at D C, and 
when at P=V. Join SP, FP. Then (Art. 174,) the velocity at 
D is to the velocity at P, as SG to SK ; that is, 

C 2 : V 2 : : SG 2 : DC 2 . 

But because the triangles SGP, FHP, are equiangular, having 
right angles at G and H, and also, from the nature of the ellipse, 
the angles SPG, FPH, equal, 

SP : PF : : SG : FH : : SG 2 : CD 2 =FHxSG 
.-. SP:PF::C 2 :V 2 

178. If of two bodies gravitating to the same center, one descends 
in a straight line, and the other revolves in a curve ; then, if the ve- 
locities of these bodies are equal in any one case, when they are 



Fig. 38. 

equally distant from the center, they will always be equal when they 
are equally distant from it. 

Let ABC (Fig. 38,) be a curve which a body 
describes about a center S to which it gravi- 
tates, while another body descends in a 
straight line AS to that center. Let BC be 
any arc of the curve ABC, and let BD, CH, 
be arcs of circles described from the center 
S, intersecting the line AS in D and H. 
From the center S describe the arc bd, in- 
definitely near to BD, and draw E/ perpen- 
dicular to ~Bb. Then, because the distances 
SD and SB are equal, the forces of gravity 
at D and B are also equal. Let these forces 
be expressed by the equal lines Dd and BE ; 
and let the force BE be resolved into the 
forces E/" and B/*. The force E/, acting at 
right angles to the path of the body, will not affect its velocity in 
that path, but will only draw it aside from a rectilinear course and 
make it proceed in the curve B&C. But the other force B/", acting 
in the direction of the course of the body, will be wholly employed 
in accelerating it. And because B and b are indefinitely near to 
each other, and likewise D and d, the accelerating force from B to 
b and from D to d, may be considered as acting uniformly. 
Therefore, the accelerations of the bodies in D and B, produced 
in equal times, are as the lines Dd, B/"; and hence, putting d for 
the increment of velocity at d, and/ for the increment of velocity 

d:f:i Dd or BE :B/. (1) 

And because the angle at E is a right angle, 

Hence, BE : B/: : v/B6 : x/B/ (2) 

And, (1) and (2), d :/: : x/B6 : x/B/ (3) 

But, putting 6 for the velocity at &, and observing that, in falling 
bodies, the velocities are as the square roots of the spaces, 
bifn x/B6: VB/"- (4) 

Therefore, (3) and (4), b:f::d:f.: b=d ; that is, the velocity at 
b equals the velocity at d. And, since the same reasoning holds 


for successive points that may be taken at equal distances from B 
and D, therefore, if of two bodies, &c.* 

179. The law according to which the planets gravitate is such, that 
any body under the influence of the same force, and falling direct to 
the sun, will have its velocity at any point equal to a constant velocity 
multiplied into the- square root of the distance it has fallen through, 
divided by the square root of the distance between the body and the 
mrfs center. 

Suppose a planet to revolve in the elliptical orbit APB (Fig. 37); 

at A, the higher apsis, the velocity V=C ~P (Art. 177) ; or 

if AN, in the axis produced=AF, v=c Let a body at 

A begin to descend towards S with this velocity, then if SL=SP, 
the velocity of the planet at P will be the same as that of the fall- 
ing body at L, (Art. 178.) But the velocity of the planet at P is 

\PS/ ~^ (sT~) ^ Ut ^ verity * s e q ua l to the constant ve- 
locity expressed by C, multiplied into the square root of NL, the 
distance fallen through,J divided by the square root of LS, the 
distance between the body and the sun's center. 

180. The force with which any planet gravitates to the sun, is in- 
versely as the square of its distance from the sun's center. 

Let C (Fig. 39,) be the center to which the falling body gravi- 
tates, A the point from which it begins to fall, and its velocity at 
any point B, is to its velocity in the point G, which bisects AC, as 

(BC/ : L " Let DEF be a curve such that if AD be an ordinate 

or a perpendicular to AC, meeting the curve in D, and BE any other 

* Principia, Lib. i, Pr. 40. Stewart's Math, and Phys. Essays, Pr. 13. 
t For SN=AB=SP-j-PF=:SP-[-NL .-. PF=NL. 

* That NL (=PF) is the distance fallen through to acquire the velocity at P, is de- 
monstrated by writers on Central Forces. (See Vince, Syst. Ast., Art. 823.) 

Playfair, Phys. Ast. 

|| For, denoting the velocity at B by V, and the velocity at G by V, 



ordinate, AD is to BE as the force at A to the force at B, then 
will twice the area ABED be equal to the Fig. 39 

square of the velocity which the body has 
acquired in B.* If therefore the velocity 33 
at B be V, that at the middle point G being c, l 

=c (?)*, and therefore 2 ABED-c 2 . ? ; 

and since AB =AC - BC, 2 ABED = c>. 

AC-BC 9 /AC \ 

- & I - 1 1. For the same reason, 
BO \BC / 

if be be drawn indefinitely near to BE, 2AbeD 
* \ f~* \ 

=c 2 ( 1 ), and therefore the difference of 
V 0C / 

these areas, or 2RbeE, that is, 


AC\ AC(BC-6C) 2 ACxB6 , xr , 
-BcH BCX6C =C *- WhereforMmdmgby 

; now c 2 and AG are constant 

quantities, therefore EB varies inversely as BC 2 . But EB repre- 
sents the force of gravity at B, and BC the distance from the 
sun. Therefore, the force of gravity of a planet in different parts 
of its orbit, is inversely as the square of its distance from the sun. 

181. The line CG is the same with the mean distance of the 
planet in an orbit of which AC is the length of the transverse axis ; 
and if the gravitation at that distance =F, and the mean distance 

itself=c, then since EB-c 2 -, F=c 2 x-=-, or F=c 2 . 

* a 

* This principle is demonstrated by the aid of Fluxions as follows : 

By construction, BE is proportional to the force at B=^-, v being the velocity 

which the moving body has acquired at B, and t the time of the descent from A to B. 
Now B6 is the momentary increment of BA the space, and therefore=ud* ; therefore 
BExB6=wd0. And 2BExB&r=2t)d. But BExB6 is the momentary increment of 
the area ABED, and %vdv is the momentary increment of t) 2 ; therefore the square of 
the velocity of the moving body, and twice the area of ABED, increase at the same 
rate, and begin to exist at the same time ; therefore they are equal. (See Playfair'a 
Outlines, Mechanics, Art. 96.) 

t iC being ultimately equal to BC. 


182. The squares of the times of revolution of any two planets, 
are as the cubes of their mean distances from the sun. 

If a be the mean distance, or the semi-transverse axis, b the 
semi-conjugate, then #a&=area of the orbit.* But as c is the ve- 
locity at the mean distance, or the elliptic arch which the planet 
moves over in a second when it is at D, (Fig. 37.) the vertex of the 
conjugate axis, therefore ^bc is the area described in that second 
by the radius vector ; and since the area is the same for every 
second of the planet's revolution (Art. 172,) therefore the area of 
the orbit divided by %bc will give the number of seconds in 

which the revolution is completed, which= - = - ; or, since 

\bc c 

c* = aF, (Art. 181,) the time of a revolution = ==%<K\/ =. 


Hence, let t, t', be the times of revolutions for two different plan- 
ets, of which the mean distances are a, a', and the force of gravity 

at those distances F, P. Then titfiiZ* \/- - 2 *\/Y> : : 

?s-, or $ : t' z : : a 3 : a' 3 . That is, the squares of the times are as the 

cubes of the mean distances ; or, since the major axes of the or- 
bits are double the mean distances, the squares of the times are as 
the cubes of the major axes. 

183. This is one of Kepler 's three great Laws, which, taken in 
connexion, are as follows : 

1. The orbits of all the planets are ellipses, the sun occupying the 
common focus. (Art. 176.) 

2. The radius vector of any planet describes areas proportional 
to the times. (Art. 172.) 

3. The squares of the periodical times are as the cubes of the ma- 
jor axes of the orbits. (Art. 182.) 

These great and fundamental principles of the planetary mo- 
tions, were discovered by the illustrious Kepler by long and as- 
siduous study of the observations made by Tycho Brahe, and 

* Day's Mensuration. 


hence he has been called the legislator of the skies. They, there 
fore, became known as facts, before they were demonstrated 
mathematically. The glory of this achievement was reserved 
for Newton, who proved that they were necessary results of the 
law of universal gravitation. 


184. Having now acquired some knowledge of the law of uni- 
versal gravitation, let us next endeavor to gain a just conception 
of the forces by which the planets are made to revolve in their 
orbits about the sun. In obedience to the first law of motion, 
every moving body tends to move in a straight line ; and were not 
the planets deflected continually towards the sun by the force of 
attraction, these bodies as well as others would move forward in 
a rectilineal direction. We call the force by which they tend to 
such a direction the projectile force, because its effects are the 
same as though the body were originally projected from a certain 
point in a certain direction. It is an interesting problem for me- 
chanics to solve, what was the nature of the impulse originally 
given to the earth, in order to impress upon it its two motions, the 
one around its own axis, the other around the sun ? If struck in 
the direction of its center of gravity it might receive a forward 
motion, but no rotation on its axis. It must, therefore, have been 
impelled by a force, whose direction did not pass through its cen- 
ter of gravity. Bernouilli, a celebrated mathematician, has calcu- 
lated that the impulse must have been given very nearly in the 
direction of the center, the point of projection being only the 165th 
part of the earth's radius from the center.* This impulse alone 
would cause the earth to move in a right line : gravitation towards 
the sun causes it to describe an orbit. Thus a top spinning on a 
smooth plane, as that of glass or ice, if impelled in a direc- 
tion not passing through the center of gravity, may be made to 
imitate the two motions of the earth, especially if the experiment 
is tried in a concave surface like that of a large bowl. The re- 
sistance occasioned by the surface on which the top moves, and 

* Francceur, Uran. p. 49 . 



that of the air, will generally destroy the force of projection and 
cause the top to revolve in a smaller and smaller orbit ; but the 
earth meets with no such resistance, and therefore makes both her 
days and years of the same length from age to age. A body, 
therefore, revolving in an orbit about a center of attraction, is 
constantly under the influence of two forces, the projectile force, 
which tends to carry it forward in a straight line which is a tan- 
gent to its orbit, and the centripetal force, by which it tends to- 
wards the center. 

185. The most simple example we have of the combined action 
of these two forces is the motion of a missile thrown from the 
hand, or of a ball fired from a cannon. It is well known that the 
particular form of the curve described by the projectile, in either 
case, will depend upon the velocity with which it is thrown. In 
each case the body will begin to move in the line of direction in 
which it is projected, but it will soon be deflected from that line 
towards the earth. It will however continue nearer to the line of 
projection as the velocity of projection is greater. Thus let AB 

Fig. 40. 


(Fig. 40,) perpendicular to AC represent the line of projection. 
The body will, in every case, commence its motion in the line AB, 
which will therefore be the tangent to the curve it describes ; but 
if it be thrown with a small velocity, it will soon depart from the 
tangent, describing the line AD ; with a greater velocity it will 
describe a curve nearer to the tangent, as AE ; and with a still 
greater velocity it will describe the curve AF. 

As an example of a body revolving in an orbit under the influ- 
ence of two forces, suppose a body placed at any point P (Fig. 40') 
above the surface of the earth, and let PA be the direction of the 
earth's center ; that is, a line perpendicular to the horizon. If the 



body were allowed to move without receiving any impulse, it 
would descend to the earth in the direction PA with an accelerated 
motion. But suppose that, at the moment of its departure from 
P, it receives a blow in the direction PB, which would carry it to 
B in the time the body would fall from P to A ; then, under the in- 
fluence of both forces, it would descend along the curve PD. If 
a stronger blow were given to it in the direction PB, it would de- 
scribe a larger curve, PE ; or, finally, if the impulse were suffi- 
ciently strong, it would circulate quite around the earth, and re- 
turn again to P, describing the circle PFG. With a velocity of 
projection still greater, it would describe an ellipse, PIK ; and if 
the velocity were increased to a certain degree, the figure would 
become a parabola or hyperbola LMP, and never return into 

186. In figure 41, suppose the planet to have passed the point C 
with so small a velocity, that the attraction of the sun bends its 
path very much, and causes it immediately to begin to approach 
towards the sun ; the sun's attraction will increase its velocity as 
it moves through D, E, and F. For the sun's attractive force on 
the planet, when at D, is acting in the direction DS, and, on account 
of the small inclination of DE to DS, the force acting in the line 
DS helps the planet forward in the path DE, and thus increases 
its velocity. In like manner the velocity of the planet will be con- 
tinually increasing as it passes through D, E, and F ; and though 
the attractive force, on account of the planet's nearness, is so muca 
increased, and tends, therefore, to make the orbit more curved, 



yet the velocity is also so much increased, that the orbit is not 
more curved than before. The same increase of velocity occa- 
sioned by the planet's approach to the sun, produces a greater in- 
crease of centrifugal force which carries it off again. We may 
see also why, when the planet has Fig. 41. 

reached the most distant parts of its 
orbit, it does not entirely fly off, and 
never return to the sun. For when 
the planet passes along H, K, A, the 
sun's attraction retards the planet, 
just as gravity retards a ball rolled up 
hill ; and when it has reached C, its 
velocity is very small, and the attrac- 
tion at the center of force causes a 
a great deflection from the tangent, 
sufficient to give its orbit a great cur- 
vature, and the planet turns about, returns to the sun, and goes 
over the same orbit again.* As the planet recedes from the sun, 
its centrifugal force diminishes faster than the force of gravity, so 
that the latter finally preponderates^ 

187. We may imitate the motion of a body in its orbit by sus- 
pending a small ball from the ceiling by a long string. The ball 
being drawn out of its place of rest, (which is directly under the 
point of suspension,) it will tend constantly towards the ^ame 
place by a force which corresponds to the force of attraction of a 
central body. If an assistant stands under the point of suspen- 
sion, his head occupying the place of the ball when at rest, the 
ball may be made to revolve about his head as the earth or any 
planet revolves about the sun. By projecting the ball in different 
directions, and with different degrees of velocity, we may make 
it describe different orbits, exemplifying principles which have 
been explained in the foregoing propositions. 


t The centrifugal force varies inversely as the cube of the distance, while the forco 
of gravity is inversely as the square. The centrifugal force, therefore, increases faster 
than the force of gravity as a body is approaching the sun, and decreases faster as the 
body recedes from the sun. (See M. Stewart's Phys. and Math. Tracts, Prop. 8.") 




188. THE PRECESSION OF THE EQUINOXES, is a slow but continual 
shifting of the equinoctial points from east to west. 

Suppose that we mark the exact place in the heavens, where, 
during the present year, the sun crosses the equator, and that this 
point is close to a certain star ; next year the sun will cross the 
equator a little way westward of that star, and so every year a 
little further westward, until, in a long course of ages, the place 
of the equinox will occupy successively every part of the ecliptic, 
until we come round to the same star again. As, therefore, the 
sun, revolving from west to east in his apparent orbit, comes 
round towards the point where it left the equinox, it meets the 
equinox before it reaches that point. The appearance is as though 
the equinox goes forward to meet the sun, and hence the phenom- 
enon is called the Precession of the Equinoxes, and the fact is 
expressed by saying that the equinoxes retrograde on the ecliptic, 
until the line of the equinoxes makes a complete revolution from 
east to west. The equator is conceived as sliding westward on 
the ecliptic, always preserving the same inclination to it, as a ring 
placed at a small angle with another of nearly the same size, 
which remains fixed, may be slid quite around it, giving a cor- 
responding motion to the two points of intersection. It must be 
observed, however, that this mode of conceiving of the precession 
of the equinoxes is purely imaginary, and- is employed merely for 
the convenience of representation. 

189. The amount of precession annually is 50." 1 ; whence, 
since there are 3600" in a degree, and 360 in the whole circum- 
ference, and consequently, 1296000", this sum divided by 50.1 
gives 25868 years for the period of a complete revolution of the 


190. Suppose now we fix to the center of each of the two 
rings (Art. 188) a wire representing its axis, one corresponding to 
the axis of the ecliptic, the other to that of the equator, the ex- 
tremity of each being the pole of its circle. As the ring deno- 
ting the equator turns round on the ecliptic, which with its axis 
remains fixed, it is easy to conceive that the axis of the equator 
revolves around that of the ecliptic, and the pole of the equator 
around the pole of the ecliptic, and constantly at a distance equal 
to the inclination of the two circles. To transfer our conceptions 
to the celestial sphere, we may easily see that the axis of the diur- 
nal sphere, (that of the earth produced, Art. 28,) would not have 
its pole constantly in the same place among the stars, but that this 
pole would perform a slow revolution around the pole of the 
ecliptic from east to west, completing the circuit in about 26,000 
years. Hence the star which we now call the pole star, has not 
always enjoyed that distinction, nor will it always enjoy it here- 
after. When the earliest catalogues of the stars were made, this 
star was 12 from the pole. It is now 1 24', and will approach 
still nearer ; or, to speak more accurately, the pole will come still 
nearer to this star, after which it will leave it, and successively 
pass by others. In about 13,000 years, the bright star Lyra, 
which lies on the circle of revolution opposite to the present pole 
star, will be within 5 of the pole, and will constitute the Pole 
Star. As Lyra now passes near our zenith, the learner might 
suppose that the change of position of the pole among the stars, 
would be attended with a change of altitude of the north pole 
above the horizon. This mistaken idea is one of the many mis- 
apprehensions which result from the habit of considering the 
horizon as a fixed circle in space. However the pole might shift 
its position in space, we should still be at the same distance from 
it, and our horizon would always reach the same distance be- 
yond it. 

191. The precession of the equinoxes is an effect of the spheroidal 
figure of the earth, and arises from the attraction of the sun and 
moon upon the excess of matter about the earths equator. 

Were the earth a perfect sphere the attractions of the sun and 
moon upon the earth would be in equilibrium among themselves, 




But if a globe were cut out of the earth, (taking half the polar 
diameter for radius,) it would leave a protuberant mass of matter 
in the equatorial regions, which may be considered as all collected 
into a ring resting on the earth. The sun being in the ecliptic, 
while the plane of this ring is inclined to the ecliptic 23 28', of 
course the action of the sun is oblique to the ring, and may be 
resolved into two forces, one in the plane of the equator, and the 
other perpendicular to it. The latter only can act as a disturbing 
force, and tending as it does to draw down the ring to the ecliptic, 
the ring would turn upon the line of the equinoxes as upon a 
hinge, and dragging the earth along with it, the equator would 
ultimately coincide with the ecliptic were it not for the revolution 
of the earth upon its axis. This may be better understood by the 
aid of a diagram. Let TAB (Fig. 42,) represent the equator, 

Fig. 42. 

TED the ecliptic, and AD the solstitial colure. Let AB be the 
movement of rotation for a very short time, being of course in the 
order of the signs and in the direction of the equator. Let BC be 
the movement produced by the disturbing force of the sun in the 
same time. The point A will describe the diagonal AC, the equa- 
tor will take the inclined situation CAT' ; the equinoctial point 
will retrograde from T to T' ; the colure AD will take the posi- 
tion AE, while the inclination of the two planes, that is, the ob- 
liquity of the ecliptic, will remain nearly the same.* 

192. The moon conspires with the sun in producing the pre- 
cession of the equinoxes, its effect, on account of its nearness to 
the earth, being more than double that of the sun, or as 7 to 3. 
The planets likewise, by their attraction, produce a small effec 

Delambre, t. 3, p. 145. Playfair's Outlines, 2, 308. 


upon the equatorial ring, but the result is slightly to diminish the 
amount of precession. The whole effect of the sun and moon 
being 50."41, that of the planets is 0.31, leaving the actual amount 
of precession 50."!.* 

This effect is not to be imagined as taking place merely at the 
time of the equinoxes, but as resulting constantly from the action 
of the sun and moon on the equatorial ring, and at every revolu- 
tion of this ring along with the earth on its axis. Conceive of 
any point in the ring, and follow it round in the diurnal revolution, 
and it will be seen that that point, in consequence of the attrac- 
tion of the sun and moon, will be made to cross the ecliptic a little 
further westward than on the preceding day. 

193. The time occupied by the sun in passing from the equinoc- 
tial point round to the same point again, is called the TROPICAL YEAR. 
As the sun does not perform a complete revolution in this inter- 
val, but falls short of it 50." 1, the tropical year is shorter than the 
sidereal by 20m. 20s. in mean solar time, this being the time of 
describing an arc of 50." 1 in the annual revolution.! The 
changes produced by the precession of the equinoxes in the ap- 
parent places of the circumpolar stars, have led to some interest- 
ing results in chronology. In consequence of the retrograde mo- 
tion of the equinoctial points, the signs of the ecliptic (Art. 35,) 
do not correspond at present to the constellations which bear the 
same names, but lie about one whole sign or 30 westward of 
them. Thus, that division of the ecliptic which is called the sign 
Taurus, lies in the constellation Aries, and the sign Gemini in the 
constellation Taurus. Undoubtedly, however, when the ecliptic 
was thus first divided, and the divisions named, the several con- 
stellations lay in the respective divisions which bear their names. 
How long is it, then, since our zodiac was formed ? 

50."1 : 1 year :: 30(=108000") : 2155.6 years. 

The result indicates that the present divisions of the zodiac 
were made soon after the establishment of the Alexandrian school 
of astronomy. (Art 2.) 

* Francoeur, Uran. 162. t 59' 8."3 : 24h. : : 50."1 : 20m. 20s. 

108 THE SUN. 


194. NUTATION is a vibratory motion of the earth's axis, arising 
from periodical fluctuations in the obliquity of the ecliptic. 

If the sun and moon moved in the plane of the equator, there 
would be no precession, and the effect of their action in producing 
it varies with their distance from that plane. Twice a year, there- 
fore, namely, at the equinoxes, the effect of the sun is nothing ; 
while at the solstices the effect of the sun is a maximum. On 
this account, the obliquity of the ecliptic is subject to a semi-an- 
nual variation, since the sun's force which tends to produce a 
change in the obliquity is variable, while the diurnal motion of 
the earth which prevents the change from taking place, is con- 
stant. Hence the plane of the equator is subject to an irregular 
motion which is called the Solar Nutation. The name is derived 
from the oscillatory motion communicated by it to the earth's axis, 
while the pole of the equator is performing its revolution around 
the pole of the ecliptic (Art. 190.) The effect of the sun however 
is less than that of the moon, in the ratio of 2 to 5. By the nuta- 
tion alone the pole of the earth would perform a revolution in a 
very small ellipse, only 18" in diameter, the center being in the 
circle which the pole describes around the pole of the ecliptic ; 
but the combined effects of precession and nutation convert the 
circumference of this circle into a wavy line. The motion of the 
equator occasioned by nutation, causes it alternately to approach 
to and recede from the stars, and thus to change their declinations. 
The solar nutation, depending on the position of the sun with re- 
spect to the equinoxes, passes through all its variations annually ; 
but the lunar nutation depending on the position of the moon with 
respect to her nodes, varies through a period of about 18 years. 


195. ABERRATION is an apparent change of place in the stars, 
occasioned by the joint effects of the motion of the earth in its orbit, 
and the progressive motion of light. 

Let EE' (Fig. 43,) represent a part of the earth's orbit, and SE 
a ray of light from the star S. Take EC and EA proportional 


to the velocity of each respectively ; com- 
plete the parallelogram, and draw the diagonal 
EB. Since an object always appears in the 
direction in which a ray of light coming from 
it, meets the eye, the combination of the two 
motions produces an impression on the eye 
exactly similar to that which would have been 
produced if the eye had remained at rest in 
the point E, and the particle of light had come 
down to it in the direction S'E ; the star, 
therefore, whose place is at S, will appear to 
the spectator at E to be situated at S'. The 
difference between its true and its apparent place, that is, the 
angle SES' is the aberration, the magnitude of which is obtained 
from the known ratio of EA to EC, or the velocity of light to that 
of the earth in its orbit. 

The velocity of light is 192,000 miles per second, while that of 
the earth in its orbit is about 19 miles per second. Represent- 
ing the velocity of light by the line EA, and that of the earth by 
AB, then, 

192,000 : 19: :Rad. : tan. 20."5=the angle at E, which is the 
amount of aberration when the direction of the ray of light is per- 
pendicular to the earth's motion. 

The effect of aberration upon the places of the fixed stars is to 
carry their apparent places a little forward of their real places in 
the direction of the earth's motion. The effect upon each particu- 
lar star will be to make it describe a small ellipse in the heavens, 
having for its center the point in which the star would be seen if 
the earth were at rest. 


196. The two points of the ecliptic where the earth is at the 
greatest and least distances from the sun respectively, do not 
always maintain the same places among the signs, but gradually 
shift their positions from west to east. If we accurately observe 
the place among the stars, where the earth is at the time of its 
perihelion the present year, we shall find that it will not be pre- 

110 THE SUN. 

cisely at that point the next year when it arrives at its perihelion, 
but about 12" (ll."66) to the east of it. And since the equinox 
itself, from which longitude is reckoned, moves in the opposite 
direction 50." 1 annually, the longitude of the perihelion increases 
every year 61."76, or a little more than one minute. This fact 
is expressed by saying that the line of the apsides of the earth's 
orbit has a slow motion from west to east. It completes one entire 
revolution in its own plane in about 100,000 years (111,149.) 

The mean longitude of the perihelion at the commencement of 
the present century was 99 30' 5", and of course in the ninth 
degree of Cancer, a little 'past the winter solstice. In the year 
1248, the perihelion was at the place of this solstice ; and since the 
increase of longitude is 61. "76 a year, hence, 

61."76 : 1 : : 90 : 5246=the time occupied in passing from the 
first of Aries to the solstice. Hence, 52461248=3998, which is 
the time before the Christian era, when the perigee was at the 
first of Aries. But this differs only 6 years from the time of the 
creation of the world, which is fixed by chronologists at 4004 
years A. C. At the period of the creation, therefore, the line of 
the apsides of the earth's orbit, coincided with the line of the 

197. The angular distance of a body from its aphelion is called 
its Anomaly ; and the interval between the sun's passing the point 
of the ecliptic corresponding to the earth's aphelion, and return- 
ing to the same point again, is called the anomalistic year. This 
period must be a little longer than. the sidereal year, since, in order 
to complete the anomalistic revolution, the sun must traverse an 
arc of 11. "66 in addition to 360. 

Now 360 : 365.256 : : ll."66 : 4m. 44s. 

198. Since the points of the annual orbit, where the sun is at 
the greatest and least distances from the earth, change their posi- 
tion with respect to the solstices, a slow change is occasioned in 
the duration of the respective seasons. For, let the perihelion 
correspond to the place of the winter solstice, as was the case in 
the year 1248 ; then as the sun moves more rapidly in that part 
of his orbit, the winter months will be shorter than the summer 


But, again, let the perihelion be at the summer solstice, as it will 
be in the year 6485* ; then the sun will move most rapidly 
through the summer months, and the winters will be longer than 
the summers. At present the perihelion is so near the winter 
solstice, that, the year being divided into summer and winter by 
the equinoxes, the six winter months are passed over between seven 
and eight days sooner than the summer months. 


199. The Mean Motion of any body revolving in an orbit, is 
that which it would have if, in the same time, it revolved uniformly 
in a circle. 

In surveying an irregular field, it is common first to strike out 
some regular figure, as a square or a parallelogram, by running 
long lines, and disregarding many small irregularities in the boun- 
daries of the field. By this process, we obtain an approximation 
to the contents of the field, although we have perhaps thrown out 
several small portions which belong to it, and included a number 
of others which do not belong to it. These being separately esti- 
mated and added to or substracted from our first computation, we 
obtain the true area of the field. In a similar manner, we proceed 
in finding the place of a heavenly body, which moves in an orbit 
more or less irregular. Thus we estimate the sun's distance from 
the vernal equinox for every day of the year at noon, on the 
supposition that he moves uniformly in a circular orbit : this is 
the sun's mean longitude. We then apply to this result various 
corrections for the irregularity of the sun's motions, and thus ob- 
-ain the true longitude. 

200. The corrections applied to the mean motions of a heav- 
enly body, in order to obtain its true place, are called Equations. 
Thus the elliptical form of the earth's orbit, the precession of the 
equinoxes, and the nutation of the earth's axis, severally affect 
the place of the sun in his apparent orbit, for which equations are 
applied. In a collection of Astronomical Tables, a large part of 


112 THE SUN. 

the whole are devoted to this object. They give us the amount 
of the corrections to be applied under all the circumstances and 
constantly varying relations in which the sun, moon, and earth 
are situated with respect to each other. The angular distance of 
the earth or any planet from its aphelion, on the supposition that 
it moves uniformly in a circle, is called its Mean Anomaly : its 
actual distance at the same moment in its orbit is called its True 

Thus in figure 44, let AEB represent the orbit of the earth 
having the sun in one of the foci at S. Upon AB describe the 
circle AMB. Let E be the place of the earth in its orbit, and M 
the corresponding place in the circle ; then the angle MCA is the 
mean, and ESA the true anomaly. The difference between the 

Fig. 44. 


mean and true anomaly, MCA ESA, is called the the Equation of 
the Center, being that correction which depends on the elliptical 
form of the orbit, or on the distance of the center of attraction 
from the center of the figure, that is, on the eccentricity of the 
orbit. It is much the greatest of all the corrections used in finding 
the sun's true longitude, amounting, at its maximum, to nearly two 
degrees (1 55' 26."8.) 

* In some astronomical treatises, the anomaly is reckoned from the perihelion. 




201. NEXT to the Sun, the Moon naturally claims our attention. 
The Moon is an attendant or satellite to the earth, around which 

she revolves at the distance of nearly 240,000 miles. Her mean 
horizontal parallax being 57' 09",f consequently, sin. 57' 09" : 
semi-diameter of the earth (3956.2) : : rad. : 238,545. (Art. 87.) 

The moon's apparent diameter is 31' 7", and her real diameter 
2160 miles. For, 

Rad. : 238,545: :sin. 15' 33" : 1079.6. = moon's semi-diame- 
ter. (See Fig. 26, p. 71.) 

And, since spheres are as the cubes of the diameters, the vol- 
ume of the moon is T V tnat f tne earth. Her density is nearly 
| (.615) the density of the earth, and her mass (= ? V><.615) is 
about 8*0- 

202. The moon shines by reflected light borrowed from the 
sun, and when full, exhibits a disk of silvery brightness, diversi- 
fied by extensive portions partially shaded. By the aid of the 
telescope, we see undoubted signs of a varied surface, composed 
of extensive tracts of level country, and numerous mountains and 

203. The line which separates the enlightened from the dark 
portions of the moon's disk, is called the Terminator. (See Fig. 2. 
Frontispiece.) As the terminator traverses the disk from new to 
full rnoon, it appears through the telescope exceedingly broken in 

* Selenography is a word more appropriate to a description of the raoon, but is not 
perhaps sufficiently familiarized by use. 

t Baily's Astronomical Tables. ' ' I 


114 THE MOON. 

some parts, but smooth in others, indicating that some portions of the 
lunar surface are uneven while others are level. The broken re- 
gions appear brighter than the smooth tracts. The latter have 
been taken for seas, but it is supposed with more probability that 
they are extensive plains, since they are still too uneven for the 
perfect level assumed by bodies of water. That there are moun- 
tains in the moon, is known by several distinct indications. First, 
when the moon is increasing, certain spots are illuminated sooner 
than the neighboring places, appearing like bright points beyond 
the terminator, within the dark part of the disk. (See Fig. 2. 
Frontispiece.) Secondly, after the terminator has passed over 
them, they project shadows upon the illuminated part of the disk, 
always opposite to the Sun, corresponding in shape to the form of 
the mountain, and undergoing changes in length from night to 
night, according as the sun shines upon that part of the moon 
more or less obliquely. Many individual mountains rise to a great 
height in the midst of plains, and there are several very remarka- 
ble mountainous groups, extending from a common center in long 

204. That there are also valleys in the moon, is equally evident 
The valleys are known to be truly such, particularly by the man- 
ner in which the light of the sun falls upon them, illuminating the 
part opposite to the sun while the part adjacent is dark, as is the 
case when the light of a lamp shines obliquely into a china cup. 
These valleys are often remarkably regular, and some of them 
almost perfect circles. In several instances, a circular chain of 
mountains surrounds an extensive valley, which appears nearly 
level, except that a sharp mountain sometimes rises from the cen- 
ter. The best time for observing these appearances is near the 
first quarter of the moon, when half the disk is enlightened ;* 
but in studying the lunar geography, it is expedient to observe the 
moon every evening from new to fall, or rather through her en- 
tire series of changes. 

* It is earnestly recommended to the student of astronomy, to examine the moon re- 
peatedly with the best telescope he can command, using low powers at first, for the 
sake of a better light. 


205. The various places on the moon's disk have received ap- 
propriate names. The dusky regions, being formerly supposed to 
be seas, were named accordingly ; and other remarkable places 
have each two names, one derived from some well known spot on 
the earth, and the other from some distinguished personage. Thus 
the same bright spot on the surface of the moon is called Mount 
Sinai or Tycho, and another Mount Etna or Copernicus. The 
names of individuals, however, are more used than the others. 
The frontispiece exhibits the telescopic appearance of the full 
moon. A few of the most remarkable points have the following 
names, corresponding to the numbers and letters on the map. (See 

1. Tycho, A. Mare Humorum, 

2. Kepler, B. Mare Nubium, 

3. Copernicus, C. Mare Imbrium, 

4. Aristarchus, D. Mare Nectaris, 

5. Helicon, E. Mare Tranquilitatis, 

6. Eratosthenes, F. Mare Serenitatis, 

7. Plato, G. Mare Fecunditatis, 

8. Archimedes, H. Mare Crisium. 

9. Eudoxus, 
10. Aristotle, 

206. The method of estimating the height of lunar mountains is 
as follows. 

Let ABO (Fig. 45.) be the illuminated hemisphere of the moon, 
SO a solar ray touching the moon in O, a point in the circle which 
separates the enlightened from the dark part of the moon. All the 
part ODA will be in darkness ; but if this part contains a moun- 
tain MF, so elevated that its summit M reaches to the solar ray 
SOM, the point M will be enlightened. Let E be the place of the 
observer on the earth, the moon being at any elongation from the 
sun, as measured by the angle EOS. Draw the lines EM, EO, 
and CM, C being the center of the moon ; and let FM be the 
height of the mountain. Draw ON perpendicular to EM. The 
line EO being known, and the angle OEM being measured with a 
micrometer, the value of ON, the projection of the lime OM, be- 


Fig. 45. 


comes known. Now OM= - - ; and since OENis a very 

small angle, EON may be considered as a right angle ; conse- 

quently, MON=MOE-90. Therefore OM= 



cos. (MOE 90) 

That is, the distance between the summit 

sin. MOE sin. EOS' 
of the mountain and the illuminated part of the moon's disk, is 
equal to the projected distance as measured by the micrometer, 
divided by the sine of the moon's elongation from the sun. 

Suppose the distance OM=?zCO, where n represents the frac- 
tion the part OM is of CO as determined by observation. Then, 

/. CM=CO 

/.CM-CO or FM=CO (Vl+n 2 -l) =i:-CO, neglecting the 

higher powers of n, which would be of too little value to be worth 
taking into the account. The value of n has been found in one 
case equal to T V, which gives the height of the mountain equal to 
^j the semi-diameter of the moon, that is, 3} miles. 

When the moon is exactly at quadrature, then EOM becomes a 
right angle, and the value of OM is obtained directly from actual 
measurement ; and having CO and OM, we easily obtain CM and 
of course FM. 


207. Schroeter, a German astronomer, estimated the heights of 
the lunar mountains by observations on their shadows. He made 
them in some cases as high as ^} of the semi-diameter of the 
moon, that is, about 5 miles. The same astronomer also estimates 
the depths of some of the lunar valleys at more than four miles. 
Hence it is inferred that the moon's surface is more broken and 
irregular than that of the earth, its mountains being higher and its 
valleys deeper in proportion to the size of the moon than those of 
the earth. 

208. Dr. Her&chel is supposed also to have obtained decisive 
evidence of the existence of volcanoes in the moon, not only 
from the light afforded by their fires, but also from the formation 
of new mountains by the accumulation of matter where fires had 
been seen to exist, and which remained after the fires were extinct. 

209. Some indications of an atmosphere about the moon have 
been obtained, the most decisive of which are derived from ap- 
pearances of twilight, a phenomenon that implies the presence 
of an atmosphere. Similar indications have been detected, it is 
supposed, in eclipses of the sun, denoting a transparent refracting 
medium encompassing the moon. The lunar atmosphere, how- 
ever, if any exists, is very inconsiderable in extent and density 
compared with that of the earth.* 

210. The improbability of our ever identifying artificial struc- 
tures in the moon may be inferred from the fact that a line one 
mile in length in the moon subtends an angle at the eye of only 
about one second. If, therefore, works of art were to have a suf- 
ficient horizontal extent to became visible, they can hardly be sup- 
posed to attain the necessary elevation, when we reflect that the 
height of the great pyramid of Egypt is less than the sixth part of 
a mile. 


118 THE MOON. 


211. The changes of the moon, commonly called her Phases, 
arise from different portions of her illuminated side being turned 
towards the earth at different times. When the moon is first 
seen after the setting sun, her form is that of a bright crescent, 
on the side of the disk next to the sun, while the other portions 
of the disk shine with a feeble light, reflected to the moon from 
the earth. Every night we observe the moon to be further and 
further eastward of the sun, and at the same time the crescent 
enlarges, until, when the moon has reached an elongation from 
the sun of 90, half her visible disk is enlightened, and she is 
said to be in her first quarter. The terminator, or line which 
separates the illuminated from the dark part of the moon, is con- 
vex towards the sun from the new moon to the first quarter, and 
the moon is said to be horned. The extremities of the crescent 
are called cusps. At the first quarter, the terminator becomes a 
straight line, coinciding with a diameter of the disk ; but after 
passing this point, the terminator becomes concave towards the 
sun, bounding that side of the moon by an elliptical curve, when 
the moon is said to be gibbous. When the moon arrives at the 
distance of 180 from the sun, the entire circle is illuminated, 
and the moon is full. She is then in opposition to the sun, rising 
about the time the sun sets. For a week after the full, the moon 
appears gibbous again, until, having arrived within 90 of the sun, 
she resumes the same form as at the first quarter, being then at 
her third quarter. From this time until new moon, she exhibits 
again the form of a crescent before the rising sun, until approach- 
ing her conjunction with the sun, her narrow thread of light is lost 
in the solar blaze ; and finally, at the moment of passing the sun, 
the dark side is wholly turned towards us and for some time we 
lose sight of the moon. 

The two points in the orbit corresponding to new and full moon 
respectively, are called by the common name of syzygies ; those 
which are 90 from the sun are called quadratures; and the 
points half way between the syzygies and quadratures are called 
octants. The circle which divides the enlightened from the unen 
lightened hemisphere of the moon, is called the circle of illumina 

PHASES. 119 

toon ; that wnich divides the hemisphere that is turned towards 
us from the hemisphere that is turned from us, is called the circle 
of the disk. 

212. As the moon is an opake body of a spherical figure, and 
borrows her light from the sun, it is obvious that that half only 
which is towards the sun can be illuminated. More or less of 
this side is turned towards the earth, according as the moon is at 
a greater or less elongation from the sun. The reason of the dif- 
ferent phases will be best understood from a diagram. Therefore 
let .T (Fig. 46,) represent the earth, and S the sun. Let A, B, C, 
&c., be successive positions of the moon. At A the entire dark 

Fig. 46. 

side of the moon being turned towards the earth, the disk would 
be wholly invisible. At B, the circle of the disk cuts off a small 
part of the enlightened hemisphere, which appears in the heavens 
at b, under the form of a crescent. At C, the first quarter, the 
circle of the disk cuts off half the enlightened hemisphere, and the 
moon appears dichotomized at c. In like manner it will be seen 
that the appearances presented at D, E, F, &c., must be those 
represented at d, e,f. 


213. The moon revolves around the earth from west to east, 
the entire circuit of the heavens in about 27 days. 

120 THE MOON. 

The precise law of the moon's motions in her revolution around 
the earth, is ascertained, as in the case of the sun, (Art. 155,) by 
daily observations on her meridian altitude and right ascension. 
Thence are deduced by calculation her latitude and longitude, 
from which we find, that the moon describes on the celestial 
sphere a great circle of which the earth is the center. 

The period of the moon's revolution from any point in the 
heavens round to the same point again, is called a month. A 
sidereal month is the time of the moon's passing from any star, 
until it returns to the same star again. A synodical month* is 
the time from one conjunction or new moon to another. The 
synodical month is about 29 days, or more exactly, 29d. 12h. 
44m. 2 S .8=29.53 days. The sidereal month is about two days 
shorter, being 27d. 7h. 43m. 1 l s .5=27.32 days. As the sun and 
moon are both revolving in the same direction, and the sun is 
moving nearly a degree a day, during the 27 days of the moon's 
revolution, the sun must have moved 27. Now since the moon 
passes over 360 in 27.32 days, her daily motion must be 13 17'. 
It must therefore evidently take about two days for the moon to 
overtake the sun. The difference between these two periods 
may, however, be determined with great exactness. The mid- 
dle of an eclipse of the sun marks the exact moment of conjunc- 
tion or new moon; and by dividing the interval between any 
two solar eclipses by the number of revolutions of the moon, or 
lunations, we obtain the precise period of the synodical month. 
Suppose, for example, two eclipses occur at an interval of 1,000 
lunations ; then the whole number of days and parts of a day 
that compose the interval divided by 1,000 will give the exact 
time of one lunation. f The time of the synodical month being 
ascertained, the exact period of the sidereal month may be derived 
from it. For the arc which the moon describes in order to come 
into conjunction with the sun, exceeds 360 by the space which 

* nv and oSos, implying that the two bodies come together. 

t It might at first view seem necessary to know the period of one lunation before 
we could know the number of lunations in any given interval. This period is known 
very nearly from the interval between one new moon and another. 


the sun has passed over since the preceding conjunction, that is, 
in 29.53 days. Therefore, 

365.24 : 360 :: 29.53 : 29.l=arc which the moon must de- 
scribe more than 360 in order to overtake the sun. Hence, 

13 17' : Id. ::29.l : 2.21d.=difference between the sidereal 
and synodical months; and 29.53 2.21=27.32, the time of the 
sidereal revolution. 

214. The moon's orbit is inclined to the ecliptic in an angle of 
about 5 (5 8' 48"). It crosses the ecliptic in two opposite points 
called her nodes. The amount of inclination is ascertained by 
observations on the moon's latitude when at a maximum, that 
being of course the greatest distance from the ecliptic, and there- 
fore equal to the inclination of the two circles. 

215. The moon, at the same age, crosses the meridian at differ- 
ent altitudes at different seasons of the year. The full moon, for 
example, will appear much further in the south when on the meri- 
dian at one period of the year than at another. This is owing to 
the fact that the moon's path is differently situated with respect to 
the horizon, at a given time of night at different seasons of the 
year. By taking the ecliptic on an artificial globe to represent 
the moon's path, (which is always near it, Art. 214,) and recollect- 
ing that the new moon is seen in the same part of the heavens 
with the sun, and the full moon in the opposite part of the heavens 
from the sun, we shall readily see that in the winter the new 
moons must run low because the sun does, and for a similar rea- 
son the full moons must run high. It is equally apparent that, in 
summer, when the sun runs high, the new moons must cross the 
meridian at a high, and the full moons at a low altitude. This 
arrangement gives us a great advantage in respect to the amount 
of light received from the moon ; since the full moon is longest 
above the horizon during the long nights of winter, when her pre- 
sence is most needed. This circumstance is especially favorable to 
the inhabitants of the polar regions, the moon, when full, travers- 
ing that part of her orbit which lies north of the equator, and of 
course above the horizon of the north pole, and traversing the por- 
tion that lies south of the equator, and below the polar horizon, 


122 THE MOON. 

when new. During the polar winter, therefore, the moon, from 
the first to the last quarter, is commonly above the horizon, while 
the sun is absent ; whereas, during summer, while the sun is pre- 
sent, the moon is above the horizon while describing her first and 
last quadrants. 

216. About the time of the autumnal equinox, the mpon when 
near the full, rises about sunset for a number of nights in succes- 
sion ; and as this is, in England, the period of harvest, the phe- 
nomenon is called the Harvest Moon. To understand the reason 
of this, since the moon is never far from the ecliptic, we will 
suppose her progress to be in the ecliptic. If the moon moved 
in the equator, then, since this great circle is at right angles to 
the axis of the earth, all parts of it, as the earth revolves, cut the 
horizon at the same constant angle. But the moon's orbit, or 
the ecliptic, which is here taken to represent it, being oblique 
to the equator, cuts the horizon at different angles in different 
parts, as will easily be seen by reference to an artificial globe. 
When the first of Aries, or vernal equinox, is in the eastern hori- 
zon, it will be seen that the ecliptic, (and consequently the moon's 
orbit,) makes its least angle with the horizon. Now at the au- 
tumnal equinox, the sun being in Libra, the moon at the full is in 
Aries, and rises when the sun sets. On the following evening, 
although she has advanced in her orbit about 13, (Art. 213,) yet 
her progress being oblique to the horizon, and at a small angle 
with it, she will be found at this time but a little way below the 
horizon, compared with the point where she was at sunset the 
preceding evening. She therefore rises but little later, and so 
for a week only a little later each evening than she did the pre- 
ceding night. 

217. The moon is about ^ nearer to us when near the zenith 
than when in the horizon; 

The horizontal distance CD (Fig. 47,) is nearly equal to AD= 
AD', which is greater than CD' by AC, the semi-diameter of the 
earth =aV the distance of the moon. Accordingly, the apparent 
diameter of the moon, when actually measured, is about 30" 
(which equals about -gV f tne whole) greater when in the zenith 



than in the horizon. The apparent enlargement of the full moon 
when rising, is owing to the same causes as that of the sun, as ex- 
plained in article 96. 

Fig. 47. 

218. The moon turns on its axis in *he same time in which it 
revolves around the earth. 

This is known by the moon's always keeping nearly the same 
face towards us, as is indicated by the telescope, which could not 
happen unless her revolution on her axis kept pace with her mo- 
tion in her orbit. Thus, it will be seen by inspecting figure 31, 
that the earth turns different faces towards the sun at different 
times ; and if 'a ball having one hemisphere white and the other 
black be carried around a lamp, it will easily be seen that it can- 
not present the same face constantly towards the lamp unless it 
turns once on its axis while performing its revolution. The same 
thing will be observed when a man walks around a tree, keeping 
his face constantly towards it. Since however the motion of the 
moon on its axis is uniform, while the motion in its orbit is une- 
qual, the moon does in fact reveal to us a little sometimes of one 
side and sometimes of the other. Thus when the ball above 
mentioned is placed before the eye with its light side towards us, 
or carrying it round, if it is moved faster than it is turned on its 
axis, a portion of the dark hemisphere is brought into view on 
one side ; or if it is moved forward slower than it is turned on 
its axis, a portion of the dark hemisphere comes into view on the 
other side. 

219. These appearances are called the moon's librations in lon- 
gitude. The moon has also a libration in latitude, so called, be- 
cause in one part of her revolution, more of the region around one 

124 THE MOON. 

of the poles comes into view, and in another part of the revolu- 
tion, more of the region around the other pole ; which gives the ap- 
pearance of a tilting motion to the moon's axis. This has nearly the 
same cause with that which occasions our change of seasons. The 
moon's axis being inclined to that of the ecliptic, about 1 degrees, 
(1 30' 10".8,) and always remaining parallel to itself, the circle 
which divides the visible from the invisible part of the moon, will 
pass in such a way as to throw sometimes more of one pole into 
view and sometimes more of the other, as would be the case with 
the earth if seen from the sun. (See Fig. 31.) 

The moon exhibits another phenomenon of this kind called 
her diurnal libration, depending on the daily rotation of the 
spectator. She turns the same face towards the center of the 
earth only, whereas we view her from the surface. When she is 
on the meridian, we see her disk nearly as though we viewed it 
from the center of the earth, and hence in this situation it is sub- 
ject to little change ; but when near the horizon, our circle of 
vision takes in more of the upper limb than would be presented 
to a spectator at the center of the earth. Hence, from this cause, 
we see a portion of one limb while the moon is rising, which is 
gradually lost sight of, and we see a portion of the opposite limb 
as the moon declines towards the west. It will be remarked that 
neither of the foregoing changes implies any actual motion in the 
moon, but that each arises from a change of position in the spec- 
tator relative to the moon. 

220. An inhabitant of the moon would have but one day and 
one night during the whole lunar month of 29 days. One of 
its days, therefore, is equal to nearly 15 of ours. So protracted 
an exposure to the sun's rays, especially in the equatorial regions 
of the moon, must occasion an excessive accumulation of heat ; 
and so long an absence of the sun must occasion a corresponding 
degree of cold. Each day would be a wearisome summer ; each 
night a severe winter.* A spectator on the side of the moon 
which is opposite to us would never see the earth ; but one on the 
side next to us would see the earth presenting a gradual succession 

* Francceur, Uranog. p. 91. 


of changes during his long night of 360 hours. Soon after the 
earth's conjunction with the sun, he would have the light of the 
earth reflected to him, presenting at first a crescent, but enlarging, 
as the earth approaches its opposition, to a great orb, 13 times as 
large as the full moon appears to us, and affording nearly 1 3 times 
as much light. Our seas, our plains, our mountains, our volcanoes, 
and our clouds, would produce very diversified appearances, as 
would the various parts of the earth brought successively into 
view by its diurnal rotation. The earth while in view to an in- 
habitant of the moon, would remain immovably fixed in the same 
part of the heavens. For being unconscious of his own motion 
around the earth, as we are of our motion around the sun, the 
earth would seem to revolve around his own planet from west to 
east ; but, meanwhile, his rotation along ^vith the moon on her 
axis, would cause the earth to have an apparent motion westward 
at the same rate. The two motions, therefore, would exactly 
balance each other, and the earth would appear all the while at 
rest. The earth is full to the moon when the latter is new to us ; 
and universally the two phases are complementary to each other.* 

221. It has been observed already, (Art. 214,) that the moon's 
orbit crosses the ecliptic in two opposite points called the nodes. 
That which the moon crosses from south to north, is called the 
ascending node ; that which the moon crosses from north to south, 
the descending node. 

From the manner in which the figure representing the earth's 
orbit and that of the moon, is commonly drawn, the learner is 
sometimes puzzled to see how the orbit of the moon can cut the 
ecliptic in two points directly opposite to each other. But he must 
reflect that the lunar orbit cuts the plane of the ecliptic and not 
the earth's path in that plane, although these respective points are 
projected upon that path in the heavens. 

222. We have thus far contemplated the revolution of the moon 
around the earth as though the earth were at rest. But, in order 
to have just ideas respecting the moon's motions, we must recol- 
lect that the moon likewise revolves along with the earth around 

* Francoeur, p. 92. 

126 THE MOON. 

the sun. It is sometimes said that the earth carries the moon 
along with her in her annual revolution. This language may 
convey an erroneous idea ; for the moon, as well as the earth, 
revolves around the sun under the influence of two forces, and 
would continue her motion around the sun, were the earth re- 
moved out of the way. Indeed, the moon is attracted towards 
the sun 2} times more than towards the earth,* and would aban- 
don the earth were not the latter also carried along with her by 
the same forces. So far as the sun acts equally on both bodies, 
their motion with respect to each other would not be disturbed. 
Because the gravity of the moon towards the sun is found to be 
greater, at the conjunction, than her gravity towards the earth, 
some have apprehended that, if the doctrine of universal gravi- 
tation is true, the moon ought necessarily to abandon the earth. 
In order to understand the reason why it does not do thus we 
must reflect, that when a body is revolving in its orbit under the 
action of the projectile force and gravity, whatever diminishes 
the force of gravity while that of projection remains the same, 
causes the body to recede from the center; and whatever in 
creases the amount of gravity carries the body towards the center 
Now, when the moon is in conjunction, her gravity towards the 
earth acts in opposition to that towards the sun, while her velocity 
remains too great to carry her, with what force remains, in a 
circle about the sun, and she therefore recedes from the sun, and 
commences her revolution around the earth. On arriving at the 
opposition, the gravity of the earth conspires with that of the sun, 
and the moon's projectile force being less than that required to 
make her revolve in a circular orbit, when attracted towards the 
sun by the sum of these forces, she accordingly begins to approach 
the sun and descends again to the conjunction.! 

* It is shown by writers on Mechanics, that the forces with which bodies revolving 
in circular orbits tend towards their centers, are as the radii of their orbits divided 
by the squares of their periodical times. Hence, supposing the orbits of the earth and 
the moon to be circular, (and their slight eccentricity will not much affect ihe re- 
sult,) we have 

400 1 

G : 

t M'Laurin's Discoveries of Newton, B. ;v, ch. 5. 


223. The attraction of the sun, however, being every where 
greater than that of the earth, the actual path of the moon around 
the sun is every where concave towards the latter. Still the el- 
liptical path of the moon around the earth, is to be conceived of 
in the same way as though both bodies were at rest with respect 
to the sun. Thus, while a steamboat is passing swiftly around an 
island, and a man is walking slowly around a post in the cabin, 
the line which he describes in space between the forward motion 
of the boat and his circular motion around the post, may be every 
where concave towards the island, while his path around the post 
will still be the same as though both were at rest. A nail in the 
rim of a coach wheel, will turn around the axis of the wheel, when 
the coach has a forward motion in the same manner as when the 
coach is at rest, although the line actually described by the nail 
will be the resultant of both motions, and very different from 



224. WE have hitherto regarded the moon as describing a great 
circle on the face of the sky, such being the visible orbit as seen 
by projection. But, on more exact investigation, it is found that 
her orbit is not a circle, and that her motions are subject to very 
numerous irregularities. These will be best understood in con- 
nection with the causes on which they depend. The law of uni- 
versal gravitation has been applied with wonderful success to their 
investigation, and its results have conspired with those of long 
continued observation, to furnish the means of ascertaining with 
great exactness the place of the moon in the heavens at any given 
instant of time, past or future, and thus to enable astronomers to 
determine longitudes, to calculate eclipses, and to solve various 
other problems of the highest interest. A complete understand- 
ing of all the irregularities of the moon's motions, must be sought 

128 THE MOON. 

for in more extensive treatises of astronomy than the present ; but 
some general acquaintance with the subject, clear and intelligible 
as far as it goes, may be acquired by first gaining a distinct idea 
of the mutual actions of the sun, the moon, and the earth. 

225. The irregularities of the moon's motions, are due chiefly to 
the disturbing influence of the sun, which operates in two ways ; first, 
by acting unequally on the earth and moon, and, secondly, by acting 
obliquely on the moon, on account of the inclination of her orbit to 
the ecliptic.* 

If the sun acted equally on the earth and moon, and always in 
parallel lines, this action would serve only to restrain them in their 
annual motions round the sun, and would not affect their actions 
on each other, or their motions about their common center of 
gravity. In that case, if they were allowed to fall directly to- 
wards the sun, they would fall equally, and their respective situa- 
tions would not be affected by their descending equally towards 
it. We might then conceive them as in a plane, every part of 
which being equally acted on by the sun, the whole plane would 
descend towards the sun, but the respective motions of the earth 
and the moon in this plane, would be the same as if it were qui- 
escent. Supposing then this plane and all in it, to have an annual 
motion imprinted on it, it would move regularly round the sun, 
while the earth and moon would move in it with respect to each 
other, as if the plane were at rest, without any irregularities. 
But because the moon is nearer the sun in one half of her orbit 
than the earth is, and in the other half of her orbit is at a greater 
distance than the earth from the sun, while the power of gravity 
is always greater at a less distance ; it follows, that in one half of 
her orbit the moon is more attracted than the earth towards the 
sun, and in the other half less attracted than the earth. The ex- 
cess of the attraction, in the first case, and the defect in the second, 
constitutes a disturbing force, to which we may add another, 
namely, that arising from the oblique action of the solar force, 
since this action is not directed in parallel lines, but in lines that 
meet in the center of the sun. 

* M'Laurin's Discoveries of Newton, B. iv, ch. 4. La Place's Syst. du Monde, 
B. iv, ch. 5. 



226. To see the effects of this process, let us suppose that the 
projectile motions of the earth and moon were destroyed, and 
that they were allowed to fall freely towards the sun. If the 
moon was in conjunction with the sun, or in that part of her orbit 
which is nearest to him, the moon would be more attracted than 
the earth, and fall with greater velocity towards the sun ; so that 
the distance of the moon from the earth would be increased in the 
fall. If the moon was in opposition, or in the part of her orbit 
which is furthest from the sun, she would be less attracted than 
the earth by the sun, and would fall with a less velocity towards 
the sun, and would be left behind ; so that the distance of the 
moon from the earth would be increased in this case also. If the 
moon was in one of the quarters, then the earth and moon being 
both attracted towards the center of the sun, they would both de- 
scend directly towards that center, and by approaching it, they 
would necessarily at the same time approach each other, and in 
this case their distance from each other would be diminished. 
Now whenever the action of the sun would increase their distance, 
if they were allowed to fall towards the sun, then the sun's action, 
by endeavoring to separate them, diminishes their gravity to each 
other ; whenever the sun's action would diminish the distance, then 
it increases their mutual gravitation. Hence, in the conjunction 
and opposition, that is, in the syzygies, their gravity towards each 
other is diminished by the action of the sun, while in the quadra- 
tures it is increased. But it must be remembered that it is not 
the total action of the sun on them that disturbs their motions, 
but only that part of it which tends at one time to separate them, 
and at another time to bring them nearer together. The other 
and far greater part, has no other effect than to retain them in 
their annual course around the sun. 

227. Suppose the moon setting out from the quarter that pre- 
cedes the conjunction with a velocity that would make her de- 
scribe an exact circle round the earth, if the sun's action had no 
effect on her : since her gravity is increased by that action, she must 
descend towards the earth and move within that circle. Her or- 
oit then would be more curved than it otherwise would have been; 
because the addition to her gravity will make her fall further at 




the end of an arc below the tangent drawn at the other end of it. 
Her motion will be thus accelerated, and it will continue to be 
accelerated until she arrives at the ensuing conjunction, because 
the direction of the sun's action upon her, during that time, makes 
an acute angle with the direction of her motion. (See Fig. 41.) 
At the conjunction, her gravity towards the earth being diminished 
by the action of the sun, her orbit will then be less curved, and 
she will be carried further from the earth as she moves to the next 
quarter ; and because the action of the sun makes there an obtuse 
angle with the direction of her motion, she will be retarded in the 
same degree as she was accelerated before. 

228. After this general explanation of the mode in which the 
sun acts as a disturbing force on the motions of the moon, the 
learner will be prepared to understand the mathematical develop- 
ment of the same doctrine. 

Therefore, let ADBC (Fig. 48,) be the orbit, nearly circular, in 
which the moon M revolves in the direction CADB, round the 
earth E. Let S be the sun, and let 
SE the radius of the earth's orbit, 
be taken to represent the force with 
which the earth gravitates to the sun. 

Then (Art. 180,) -^L: ^ : : SE : 

= the force by which the sun 
draws the moon in the direction 


MS. Take MG=^=,, and let the 

SM 2 

parallelogram KF be described, 
having MG for its diagonal, and 
having its sides parallel to EM and 
ES. The force MG may be re- 
solved into two, MF and MK, of 
which MF, directed towards E, the 
center of the earth, increases the 
gravity of the moon to the earth, and does not hinder the areas 
described by the radius vector from being proportional to the 


times. The other force MK draws the moon in the direction of 
the line joining the centers of the sun and earth. It is, however, 
only the excess of this force, above the force represented by SE, 
or that which draws the earth to the sun, which disturbs the rela- 
tive position of the moon and earth. This is evident, for if KM 
were just equal to ES, no disturbance of the moon relative to the 
earth could arise from it. If then ES be taken from MK, the dif- 
ference HK is the whole force in the direction parallel to SE, by 
which the sun disturbs the relative position of the moon and earth. 
Now, if in MK, MN be taken equal to HK, and if NO be drawn 
perpendicular to the radius vector EM produced, the force MN 
may be resolved into two, MO and ON, the first lessening the 
gravity of the moon to the earth ; and the second, being parallel 
to the tangent of the moon's orbit in M, accelerates the moon's 
motion from C to A, and retards it from A to D, and so alternately 
in the other two quadrants. Thus the whole solar force directed 
to the center of the earth, is composed of the two parts MF and 
MO, which are sometimes opposed to one another, but which 
never affect the uniform description of the areas about E. Near 
the quadratures the force MO vanishes, and the force MF, which 
increases the gravity of the moon to the earth, coincides with CE 
or DE. As the moon approaches the conjunction at A, the force 
MO prevails over MF, and lessens the gravity of the moon to the 
earth. In the opposite point of the orbit, when the moon is in op- 
position at B, the force with which the sun draws the moon is less 
than that with which the sun draws the earth, so that the effect of 
the solar force is to separate the moon and earth, or to increase 
their distance ; that is, it is the same as if, conceiving the earth 
not to be acted on, the sun's force drew the moon in the direction 
from E to B. This force is negative, therefore, in respect to the 
force at A, and the effect in both cases is to draw the moon from 
the earth in a direction perpendicular to the line of the quadra- 
tures. Hence, the general result is, that by the disturbing force 
of the sun, the gravity to the earth is increased at the quadratures, 
and diminished at the syzygies. It is found by calculation that the 
average amount of this disturbing force is ^ of the moon's 
gravity to the earth.* 


132 THE MOON. 

229. With these general principles in view, we may now pro- 
ceed to investigate the figure of the moon's orbit, and the irregu- 
larities to which the motions of this body are subject. 

230. The figure of the moon's orbit is an ellipse, having the earth 
in one of the foci. 

The elliptical figure of the moon's orbit, is revealed to us by ob- 
servations on her changes in apparent diameter, and in her hori- 
zontal parallax. First, we may measure from day to day the ap- 
parent diameter of the moon. Its variations being inversely as 
the distances, (Art. 163,) they give us at once the relative distance 
of each point of observation from the focus. Secondly, the va- 
riations on the moon's horizontal parallax, which also are inversely 
as the distances, (Art. 82,) lead to the same results. Observations 
on the angular velocities, combined with the changes in the lengths 
of the radius vector, afford the means of laying down a plot of the 
lunar orbit, as in the case of the sun, represented in figure 32. 
The orbit is shown to be nearly an ellipse, because it is found to 
have the properties of an ellipse. 

The moon's greatest and least apparent diameters are respectively 
33'.518 and 29'.365, while her corresponding changes of parallax 
are 61 '.4 and 53'.8. The two ratios ought to be equal, and we 
shall find such to be the fact very nearly, as expressed by the fore- 
going numbers ; for, 

61.4 : 53.8 : : 33.518 : 29.369. 

The greatest and least distances of the moon from the earth, 
derived from the parallaxes, are 63.8419 and 55.9164, or nearly 
64 and 56. the radius of the earth being taken for unity. Hence, 
taking the arithmetical mean, which is 59.879, we find that the 
mean distance of the moon from the earth is very nearly 60 times 
the radius of the earth. The point in the moon's orbit nearest 
the earth, is called her perigee ; the point furthest from the earth, 
her apogee. 

The greatest and least apparent diameters of the sun are re- 
spectively 32.583, and 31.517, which numbers express also the ratio 
of the greatest and least distances of the earth from the sun. By 
comparing this ratio with that of the distances of the moon, it will 
be seen that the latter vary much more than the former, and con- 


sequently that the lunar orbit is much more eccentric than the so- 
lar. The eccentricity of the moon's orbit is in fact 0.0548, (the 
semi-major axis being as usual taken for unity) = T 'j of its mean 
distance from the earth, while that of the earth is only .01685=^ 
of its mean distance from the sun. 

231. The moon's nodes constantly shift their positions in the eclip- 
tic from east to west, at the rate of 19 35' per annum, returning to 
the same points in 18.6 years. 

Suppose the great circle of the ecliptic marked out on the face 
of the sky in a distinct line, and let us observe, at any given time, 
the exact point where the moon crosses this line, which we will 
suppose to be close to a certain star ; then, on its next return to 
that part of the heavens, we shall find that it crosses the ecliptic 
sensibly to the westward of that star, and so on, further and fur- 
ther to the westward every time it crosses the ecliptic at either 
node. This fact is expressed by saying that the nodes retrograde 
on the ecliptic, and that the line which joins them, or the line of 
the nodes, revolves from east to west. 

232. This shifting of the moon's nodes implies that the lunar 
orbit is not a curve returning into itself, but that it more resem- 
bles a spiral like the curve represented in figure 49, where abc 
represents the ecliptic, and ABC the Fig- 49 - 

lunar orbit, having its nodes at C and 
E, instead of A and a ; consequently, 
the nodes shift backwards through 
the arcs aC and AE. The manner 
in which this effect is produced may 
be thus explained. That part of the 
solar force which is parallel to the line joining the centers of the 
sun and earth, (See Fig. 48,) is not in the plane of the moon's 
orbit, (since this is inclined to the ecliptic about 5,) except when 
the sun itself is in that plane, or when the line of the nodes being 
produced, passes through the sun. In all other cases it is oblique 
to the plane of the orbit, and may be resolved into two forces, 
one of which is at right angles to that plane, and is directed to- 
wards the ecliptic. This force of course draws the moon continu 



ally towards the ecliptic, or produces a continual deflection of the 
moon from the plane of her own orbit towards that of the earth. 
Hence the moon meets the plane of the ecliptic sooner than it 
would have done if that force had not acted. At every half revo- 
Jution, therefore, the point in which the moon meets the ecliptic, 
shifts in a direction contrary to that of the moon's motion, or con- 
trary to the order of the signs. If the earth and sun were at rest, 
the effect of the deflecting force just described, would be to pro- 
duce a retrograde motion of the line of the nodes till that line was 
brought to pass through the sun, and of consequence, the plane of 
the moon's orbit to do the same, after which they would both re- 
main in their position, there being no longer any force tending to 
produce change in either. But the motion of the earth carries the 
line of the nodes out of this position, and produces, by that means, 
its continual retrogradation. The same force produces a small 
variation in the inclination of the moon's orbit, giving it an alter- 
nate increase and decrease within very narrow limits.* These 
points will be easily understood by the aid of a diagram. There- 
fore, let MN (Fig. 50,) be the ecliptic, ANB the orbit of the moon, 
the moon being in L, and N its descending node. Let the disturb- 
ing force of the sun which tends to bring it down to the ecliptic 

Fig. 50. 

be represented by L&, and its velocity in its orbit by La. Under 
the action of these two forces, the moon will describe the diago- 
nal Lc of the parallelogram ba, and its orbit will be changed from 
AN to LN' ; the node N passes to N' ; and the exterior angle at N' 
of the triangle LNN' being greater than the interior and opposite 

* Playfair. 


angle at N, the inclination of the orbit is increased at the node. 
After the moon has passed the ecliptic to the south side to Z, the 
disturbing force Id produces a new change of the orbit N'Ze to 
N"Z/", and the inclination is diminished as at N". In general, 
while the moon is receding from one of the nodes, its inclination is 
diminishing : while it is approaching a node, the inclination is in- 

233. The period occupied by the sun in passing from one of 
the moon's nodes until it comes round to the same node again, is 
called the synodical revolution of the node. This period is shorter 
than the sidereal year, being only about 346 days. For since 
the node shifts its place to the westward 19 35' per annum, the 
sun, in his annual revolution, comes to it so much before he com- 
pletes his entire circuit ; and since the sun moves about a degree 
a day, the synodical revolution of the node is 36519=346, or 
more exactly, 346.619851. The time from one new moon, or 
from one full moon, to another, is 29.5305887 days. Now 19 
synodical revolutions of the nodes contain very nearly 223 of 
these periods. 

For 346.619851X19=6585.78, 

And 29.5305887X223=6585.32. 

Hence, if the sun and moon were to leave the moon's node toge- 
ther, after the sun had been round to the same node 19 times, the 
moon would have performed very nearly 223 synodical revolu- 
tions, and would, therefore, at the end of this period meet at the 
same node, to repeat the same circuit. And since eclipses of the 
sun and moon depend upon the relative position of the sun, the 
moon, and node, these phenomena are repeated in nearly the same 
order, in each of those periods. Hence, this period, consisting of 
about 18 years and 10 days, under the name of the Saros, was 
used by the Chaldeans and other ancient nations in predicting 

234. The Metonic Cycle is not the same with the Saros, but 
consists of 19 tropical years. During this period the moon makes 

* Francceur, Uranog. p. 158. Robison's Phys. Astronomy, Art. 264, 

136 THE MOON. 

very nearly 235 synodical revolutions, and hence the new and full 
moons, if reckoned by periods of 19 years, recur at the same 
dates. If, for example, a new moon fell on the fiftieth day of one 
cycle, it would also fall on the fiftieth day of each succeeding cycle ; 
and, since the regulation of games, feasts, and fasts, has been 
made very extensively according to new or full moons, hence this 
lunar cycle has been much used both in ancient and modern 
times. The Athenians adopted it 433 years before the Christian 
era, for the regulation of their calendar, and had it inscribed in 
letters of gold on the walls of the temple of Minerva. Hence the 
term Golden Number, which denotes the year of the lunar cycle. 

235. The line of the apsides of the moon's orbit revolves from 
west to east through her whole orbit in about nine years. 

If, in any revolution of the moon, we should accurately mark 
the place in the heavens where the moon comes to its perigee, 
(Art. 230,) we should find, that at the next revolution, it would 
come to its perigee at a point a little further eastward than before, 
and so on at every revolution, until, after 9 years, it would come 
to its perigee at nearly the same point as at first. This fact is 
expressed by saying that the perigee, and of course the apogee, 
revolves, and that the line which joins these two points, or the line 
of the apsides, also revolves. 

The place of the perigee may be found by observing when the 
moon has the greatest apparent diameter. But as the magnitude 
of the moon varies sJowly at this point, a better method of ascer- 
taining the position of the apsides, is to take two points in the or- 
bit where the variations in apparent diameter are most rapid, and 
to find where they are equal on opposite sides of the orbit. The 
middle point between the two will give the place of the perigee. 

The angular distance of the moon from her perigee in any part 
of her revolution, is called the Moon's Anomaly. 

236. The change of place in the apsides of the moon's orbit, 
like the shifting of the nodes, is caused by the disturbing influence 
of the sun. If when the moon sets out from its perigee, it were 
urged by no other force than that of projection, combined with its 
gravitation towards the earth, it would describe a symmetrical 


curve (Art. 186,) coming to its apogee at the distance of 180. 
But as the mean disturbing force in the direction of the radius 
vector tends, on the whole, to diminish the gravitation of the 
moon to the earth, the portion of the path described in an instant 
will be less deflected from her tangent, or less curved, than if this 
force did not exist. Hence the path of the moon will not inter- 
sect the radius vector at right angles, that is, she will not arrive at 
her apogee until after passing more than 180 from her perigee, 
by which means these points will constantly shift their positions 
from west to east.* The motion of the apsides is found to be 3 
1' 20" for every sidereal revolution of the moon. 

237. On account of the greater eccentricity of the moon's orbit 
above that of the sun, the Equation of the Center, or that correc- 
tion which is applied to the moon's mean anomaly to find her true 
anomaly (Art. 200,) is much greater than that of the sun, being 
when greatest more than six degrees, (6 17' 12".7,) while that of 
the sun is less than two degrees, (1 55' 26".8.) 

The irregularities in the motions of the moon may be compared 
to those of the magnetic needle. As a. first approximation, we say 
that the needle places itself in a north and south line. On closer 
examination, however, we find that, at different places, it deviates 
more or less from this line, and we introduce the first great cor- 
rection under the name of the declination of the needle. But ob- 
servation shows us that the declination alternately increases and 
diminishes every day, and therefore we apply to the decimation 
itself a second correction for the diurnal variation. Finally, we 
ascertain, from long continued observations, that the diurnal va- 
riation is affected by the change of seasons, being greater in sum- 
mer than in winter, and hence we apply to the diurnal variation a 
third correction for the annual variation. 

In like manner, we shall find the greater inequalities of the 
moon's motions are themselves subject to subordinate inequalities, 
which give rise to smaller equations, and these to smaller still, to 
the last degree of refinement. 

238. Next to the equation of the center, the greatest correction 

* Playfair. 

138 THE MOON. 

to be applied to the moon's longitude, is that which belongs to the 
Evection. The evection is a change of form in the lunar orbit, by 
which its eccentricity is sometimes increased, and sometimes 
diminished. It depends on the position of the line of the apsides 
with respect to the line of the syzygies. 

This irregularity, and its connexion with the place of the peri- 
gee with respect to the place of conjunction or opposition, was 
known as a fact to the ancient astronomers, Hipparchus and 
Ptolemy ; but its cause was first explained by Newton in con- 
formity with the law of universal gravitation. It was found, by 
observation, that the equation of the center itself was subject to a 
periodical variation, being greater than its mean, and greatest of 
all when the conjunction or opposition takes place at the perigee 
or apogee, and least of all when the conjunction or opposition 
takes place at a point half way between the perigee and apogee ; 
or, in the more common language of astronomers, the equation of 
the center is increased when the line of- the apsides is in syzygy, 
and diminished when that line is in quadrature. If, for example, 
when the line of the apsides is in syzygy, we compute the moon's 
place by deducting the equation of the center from the mean 
anomaly (see Art. 200,) seven days after conjunction, the compu- 
ted longitude will be greater than that determined by actual obser- 
vation, by about 80 minutes ; but if the same estimate is made 
when the line of the apsides is in quadrature, the computed longi- 
tude will be less than the observed, by the same quantity. These 
results plainly show a connexion between the velocity of the 
moon's motions and the position of the line of the apsides with 
respect to the line of the syzygies. 

239. Now any cause which, at the perigee, should have the 
effect to increase the moon's gravitation towards the earth beyond 
its mean, and, at the apogee, to diminish the moon's gravitation 
towards the earth, would augment the difference between the 
gravitation at the perigee and at the apogee, and consequently in- 
crease the eccentricity of the orbit. Again, any cause which at 
the perigee should have the effect to lessen the moon's gravitation 
towards the earth, and, at the apogee, to increase it, would lessen 
the difference between the two, and consequently diminish the 


eccentricity of the orbit, or bring it nearer to a circle. Let us 
see if the disturbing force of the sun produces these effects. The 
sun's disturbing force, as we have seen in article 228, admits of 
two resolutions, one in the direction of the radius vector, (OM, 
Fig. 48,) the other (ON) in the direction of a tangent to the orbit. 
First, let AB be the line of the apsides in syzygy, A being the place 
of the perigee. The sun's disturbing force OM is greatest in the 
direction of the line of the syzygies ; yet depending as it does on the 
unequal action of the sun upon the earth and the moon, and being 
greater as their distance from each other is greater, it is at a mini- 
mum when acting at the perigee, and at a maximum when acting at 
the apogee. Hence its effect is to draw away the moon from the 
earth less than usual at the perigee, and of course to make her 
gravitation towards the earth greater than usual, while at the 
apogee its effect is to diminish the tendency of the moon to the 
earth more than usual, and thus to increase the disproportion be- 
tween the two distances of the moon from the focus at these two 
points, and of course to increase the eccentricity of the orbit. 
The moon, therefore, if moving towards the perigee, is brought 
to the line of the apsides in a point between its mean place and 
the earth ; or if moving towards the apogee, she reaches the line 
of the apsides in a point more remote from the earth than its mean 

Secondly, let CD be the line of the apsides, in quadrature, C 
being the place of the perigee. The effect of the sun's disturb- 
ing force is to increase the tendency of the moon towards the 
earth when in quadrature. If, however, the moon is then at her 
perigee, such increase will be less than usual, and if at her apogee, 
it will be more than usual ; hence its effect will be to lessen the 
disproportion between the two distances of the moon from the 
focus at these two points ; and of course to diminish the eccen- 
tricity of the orbit. The moon, therefore, if moving towards 
the perigee, meets the line of the apsides in a point more remote 
from the earth than the mean place of the perigee ; and if moving 
towards the apogee, in a point between the earth and the mean place 
of the apogee.* 

* Woodhouse's Ast. p. 680. 

140 THE MOON. 

240. A third inequality in the lunar motions, is the Variation. 
By comparing the moon's place as computed from her mean mo- 
tion corrected for the equation of the center and for evection, 
with her place as determined by observation, Tycho Brahe dis- 
covered that the computed and observed places did not always 
agree. They agreed only in the syzygies and quadratures, and 
disagreed most at a point half way between these, that is, at the 
octants. Here, at the maximum, it amounted to more than half 
a degree (35' 41. "6.) It appeared evident from examining the 
daily observations while the moon is performing her revolution 
around the earth, that this inequality is connected with the angular 
distance of the moon from the sun, and in every part of the orbit 
could be correctly expressed by multiplying the maximum value 
as given above, into the sine of twice the angular distance between 
the sun and the moon. It is, therefore, at the conjunctions and 
quadratures, and greatest at the octants. Tycho Brahe knew the 
fact : Newton investigated the cause. 

It appears by article 228, that the sun's disturbing force can be 
resolved into two parts, one in the direction of the radius vector, 
the other at right angles to it in the direction of a tangent to the 
moon's orbit. The former, as already explained, produces the 
Evection: the latter produces the Variation. This latter force 
will accelerate the moon's velocity, in every point of the quadrant 
which the moon describes in moving from quadrature to conjunc- 
tion, or from C to A, (Fig. 48,) but at an unequal rate, the 
acceleration being greatest at the octant, and nothing at the quad- 
rature and the conjunction ; and when the moon is past conjunction, 
the tangential force will change its direction and retard the moon's 
motion. All these points will be understood by inspection of 
figure 48. 

241. A fourth lunar inequality is the Annual Equation. This 
depends on the distance of the earth (and of course the moon) 
from the sun. Since the disturbing influence of the sun has a 
greater effect in proportion as the sun is nearer,* consequently all 
the inequalities depending on this influence must vary at different 

* Varying reciprocally as the cube of the sun's distance from the earth. 


seasons of the year. Hence, the amount of this effect due to any 
particular time of the year is called the Annual Equation. 

242. The foregoing are the largest of the inequalities of the 
moon's motions, and may serve as specimens of the corrections that 
are to be applied to the mean place of the moon in order to find 
her true place. These were first discovered by actual observa- 
tion ; but a far greater number, though most of them are exceed- 
ingly minute, have been made known by the investigations of Phys- 
ical Astronomy, in following out all the consequences of universal 
gravitation. In the best tables, about 30 equations are applied to 
the mean motions of the moon. That is, we first compute the 
place of the moon on the supposition that she moves uniformly 
in a circle. This gives us her mean place. We then proceed, 
by the aid of the Lunar Tables, to apply the different corrections, 
such as the equation of the center, evection, variation, the annual 
equation, and so on, to the number of 28. Numerous as these 
corrections appear, yet La Place informs us, that the whole num- 
ber belonging to the moon's longitude is no less than 60 ; and 
that to give the tables all the requisite degree of precision, addi- 
tional investigations will be necessary, as extensive at least as 
those already made.* The best tables in use in the time of Tycho 
Brahe, gave the moon's place only by a distant approximation. 
The tables in use in the time of Newton, (Halley's tables,) approxi- 
mated within 7 minutes. Tables at present in use give the moon's 
place to 5 seconds. These additional degrees of accuracy have 
been attained only by immense labor, and by the united efforts of 
Physical Astronomy and the most refined observations. 

243. The inequalities of the moon's motions are divided into 
periodical and secular. Periodical inequalities are those which 
are completed in comparatively short periods, like evection and 
variation: Secular inequalities are those which are completed 
only in very long periods, such as centuries or ages. Hence the 
corresponding terms periodical equations, and secular equations. 
As an example of a secular inequality, we may mention the ac~ 

* Syst. du Monde, 1. iv,c. 5. 

142 THE MOON. 

celeration of the moon's mean motion. It is discovered, that the 
moon actually revolves around the earth in less time now than 
she did in ancient times. The difference however is exceedingly 
small, being only about 10" in a century, but increases from century 
to century as the square of the number of centuries from a given 
epoch. This remarkable fact was discovered by Dr. Halley.* In a 
lunar eclipse the moon's longitude differs from that of the sun, at the 
middle of the eclipse, by exactly 180 ; and since the sun's lon- 
gitude at any given time of the year is known, if we can learn 
the day and hour when an eclipse occurs, we shall of course know 
the longitude of the sun and moon. Now in the year 721 before 
the Christian era, on a specified day and hour, Ptolemy records a 
lunar eclipse to have happened, and to have been observed by 
the Chaldeans. The moon's longitude, therefore, for that time is 
known ; and as we know the mean motions of the moon at pre- 
sent, starting from that epoch, and computing, as may easily be 
done, the place which the moon ought to occupy at present at any 
given time, she is found to be actually nearly a degree and a half 
in advance of that place. Moreover, the same conclusion is 
derived from a comparison of the Chaldean observations with those 
made by an Arabian astronomer of the tenth century. 

This phenomenon at first led astronomers to apprehend that the 
moon encountered a resisting medium, which, by destroying at 
every revolution a small portion of her projectile force, would 
have the effect to bring her nearer and nearer to the earth and 
thus to augment her velocity. But in 1786, La Place demon- 
strated that this acceleration is one of the legitimate effects of the 
sun's disturbing force, and is so connected with changes in the 
eccentricity of the earth's orbit, that the moon will continue to be 
accelerated while that eccentricity diminishes, but when the eccen- 
tricity has reached its minimum (as it will do after many ages) 
and begins to increase, then the moon's motion will begin to be 
retarded, and thus her mean motions will oscillate forever about a 
mean value. 

244. The lunar inequalities which have been considered are such 

* Astronomer Royal of Great Britain, and cotemporary with Sir Isaac Newton. 


only as affect the moon's longitude ; but the sun's disturbing force 
also causes inequalities in the moon's latitude and parallax. Those 
of latitude alone require no less than twelve equations. Since 
the moon revolves in an orbit inclined to the ecliptic, it is easy to 
see that the oblique action of the sun must admit of a resolution 
into two forces, one of which being perpendicular to the moon's 
orbit, must effect changes in her latitude. Since also several of the 
inequalities already noticed involve changes in the length of the 
radius vector, it is obvious that the moon's parallax must be sub- 
ject to corresponding perturbations. 



245. AN eclipse of the moon happens, when the moon in its 
revolution about the earth, falls into the earth's shadow. An 
eclipse of the sun happens, when the moon, coming between the 
earth and the sun, covers either a part or the whole of the solar 
disk. An eclipse of the sun can occur only at the time of con- 
junction, or new moon ; and an eclipse of the moon, only at the 
time of opposition, or full moon. Were the moon's orbit in the 
same plane with that of the earth, or did it coincide with the 
ecliptic, then an eclipse of the sun would take place at every 
conjunction, and an eclipse of the moon at every opposition ; for 
as the sun and earth both lie in the ecliptic, the shadow of the 
earth must also extend in the same plane, being of course always 
directly opposite to the sun ; and since, as we shall soon see, the 
length of this shadow is much greater than the distance of the 
moon from the earth, the moon, if it revolved in the plane of the 
ecliptic, must pass through the shadow at every full moon. For 
similar reasons, the moon would occasion an eclipse of the sun, 
partial or total, in some portions of the earth at every new moon. 
But the lunar orbit is inclined to the ecliptic about 5, so that the 
center of the moon, when she is furthest from her node, is 5 from 

144 THE MOON. 

the axis of the earth's shadow (which is always in the ecliptic ;) 
and, as we shall show presently, the greatest distance to which the 
shadow extends on each side of the ecliptic, that is, the greatest 
semi-diameter of the shadow, where the moon passes through it, 
is only about of a degree, while the semi-diameter of the moon's 
disk is only about j of a degree ; hence the two semi-diame- 
ters, namely, that of the moon and the earth's shadow, cannot 
overlap one another, unless, at the time of new or full moon, the 
sun is at or very near the moon's node. In the course of the sun's 
apparent revolution around the earth once a year, he is succes- 
sively in every part of the ecliptic ; consequently, the conjunctions 
and oppositions of the sun and moon may occur at any part of the 
ecliptic, either when the sun is at the moon's node, (or when he 
is passing that point of the celestial vault on which the moon's 
node is projected as seen from the earth ;) or they may occur 
when the sun is 90 from the moon's node, where the lunar and 
solar orbits are at the greatest distance from each other; or, finally, 
they may occur at any intermediate point. Now the sun, in his 
annual revolution, passes each of the moon's nodes on opposite 
sides of the ecliptic, and of course at opposite seasons of the 
year ; so that, for any given year, the eclipses commonly happen 
in two opposite months, as January and July, February and 
August, May and November. These, therefore, are called Node 

246. If the sun were of the same size with the earth, the shadow 
of the earth would be cylindrical and infinite in length, since the 
tangents drawn from the sun to the earth (which form the bounda- 
ries of the shadow) would be parallel to each other ; but as the 
sun is a vastly larger body than the earth, the tangents converge 
and meet in a point at some distance behind the earth, forming a 
cone of which the earth is the base, and whose vertex (and of 
course its axis) lies in the ecliptic. A little reflection will also 
show us, that the form and dimensions of the shadow must be 
affected by several circumstances ; that the shadow must be of 
the greatest length and breadth when the sun is furthest from the 
earth ; that its figure will be slightly modified by the spheroidal 
figure of the earth ; and that the moon, being, at the time of i* 


opposition, sometimes nearer to the earth, and sometimes further 
from it, will accordingly traverse it at points where its breadth 
varies more or less. 

247. The semi-angle of the cone of the earth's shadow, is equal 
to the sun's apparent semi-diameter, minus his horizontal pat 

Let AS (Fig. 51,) be the semi-diameter of the sun, BE that of 
the earth, and EC the axis of the earth's shadow. Then the 
semi-angle of the cone of the earth's shadow ECB=AES-EAB, 

Fig. 51. 

of which AES is the sun's semi-diameter and EAB his horizontal 
parallax ; and as both these quantities are known, hence the angle 
at the vertex of the shadow becomes known. Putting <5 for the 
the sun's semi-diameter, andp for his horizontal parallax, we have 
the semi-angle of the earth's shadow ECB= p. 

248. At the mean distance of the earth from the sun, the length 
of the earth's shadow is about 860,000 miles, or more than three times 
the distance of the moon from the earth. 

In the right angled triangle ECB, right angled at B, the angle 
ECB being known, and the side EB, we can find the side EC. 


For sin. (5 p) : EB::R : EC=-^--J- . This value will vary 

sin. (d p) 

with the sun's semi-diameter, being greater as that is less. Its 
mean value being 16' 1".5 and the sun's horizontal parallax being 
9".6, 5p=I5 r 52".9, and EB=3956.2. Hence, 

Sin. 15' 53" : Rad. : : 3956.2 : 856,275. 

. Since the distance of the moon from the earth is 238,545 miles, 
the shadow extends about 3.6 times as far as the moon, and con- 


146 THE MOON. 

sequently, the moon passes the shadow towards its broadest part, 
where its breadth is much more than sufficient to cover the moon's 

249. The average breadth of the earth 1 s shadow where it eclipses 
the moon is almost three times the moon's diameter. 

Let mm' (Fig. 51,) represent a section of the earth's shadow 
where the moon passes through it, M being the center of the cir- 
cular section. Then the angle MEm will be the angular breadth 
of half the shadow. But, 

MEm = BwE BCE ; that is, since ~BmE is the moon's horizon- 
tal parallax, (Art. 82,) and BCE equals the sun's semi-diameter 
minus his horizontal parallax (&p,) therefore, putting P for the 
moon's horizontal parallax, we have 

MEm = 'P-(d-p)=P+p-5' J that is, since P=57' 1" and 
S-p=l5> 52".9, therefore, 57' I" 15' 52".9=41' 8".l, which is 
nearly three times 15' 33", the semi-diameter of the moon. Thus, 
it is seen how, by the aid of geometry, we learn to estimate vari- 
ous particulars respecting the earth's shadow, by means of simple 
data derived from observation. 

250. The distance of the moon from her node when she just 
touches the shadow of the earth, in a lunar eclipse, is called the 
Lunar Ecliptic Limit ; and her distance from the node in a solar 
eclipse, when the moon just touches the solar disk, is called the 
Solar Ecliptic Limit. The Limits are respectively the furthest 
possible distances from the node at which eclipses can take place. 

251. The Lunar Ecliptic Limit is nearly 12 degrees. 

Let CN (Fig. 52,) be the sun's path, MN the moon's, and N the 
node. Let Ca be the semi-diameter of the earth's shadow, and 
Ma the semi-diameter of the moon. Since Ca and Ma are known 

Fig. 52. 


quantities, their sum CM is also known. The angle at N is 
known, being the inclination of the lunar orbit to the ecliptic. 
Hence, in the spherical triangle MCN, right angled at M,* by 
Napier's theorem, (Art. 132, Note,) 

Rad.xsin. CM=sin. CNxsin. MNC. 

The greatest apparent semi-diameter of the earth's shadow 
where the moon crosses it, computed by article 249, is 45' 52", 
and the moon's greatest apparent semi-diameter, is 16' 45".5, 
which together, give MC equal to 62' 37". 5. Taking the incli- 
nation of the moon's orbit, or the angle MNC (what it generally 
is in these circumstances) at 5 17', and we have Rad.xsin. 

62' 37".5=sin. CNxsin. 5 17', or sin. CN- Rad - ^^f 7 ''- 8 . 

and CN^ll 25' 40". f This is the greatest distance of the moon from 
her node at which an eclipse of the moon can take place. By 
varying the value of CM, corresponding to variations in the dis- 
tances of the sun and moon from the earth, it is found that if NC 
is less than 9, there must be an eclipse ; but between this and the 
limit, the case is doubtful. 

When the moon's disk only comes in contact with the earth's 
shadow, as in figure 52 , the phenomenon is called an appulse, 
when only a part of the disk enters the shadow, the eclipse is 
said to be partial, and Mai if the whole of the disk enters the 
the shadow. The eclipse is called central when the moon's center 
coincides with the axis of the shadow, which happens when the. 
moon at the time of the eclipse is exactly at her node. 

252. Before the moon enters the earth's shadow, the earth be- 
gins to intercept from it portions of the sun's light, gradually in- 
creasing until the moon reaches the shadow. This partial light is 
called the moon's Penumbra. Its limits are ascertained by drawing 
the tangents AC'B' and A'C'B. (Fig. 51.) Throughout the space 
included between these tangents more or less of the sun's light is 
intercepted from the moon by the interposition of the earth ; for 

* The line CM is to be regarded as the projection of the line which connects the 
centers of the moon and section of the earth's shadow, as seen from the earth, 
t Woodhouse's Astronomy, p. 718. 

148 THE MOON. 

it is evident, that as the moon moves towards the shadow, she 
would gradually lose the view of the sun, until, on entering the 
shadow, the sun would be entirely hidden from her. 

253. The semi-angle of the Penumbra equals the sun's semi- 
diameter and horizontal parallax, or 5+p. 

The angle 7*C'M (Fig. 51,)=AC'S=AES+B'AE. But AES is 
the sun's semi-diameter, and B'AE is the sun's horizontal parallax, 
both of which quantities are known. 

254. The semi-angle of a section of the Penumbra, where the 
moon crosses it, equals the moon's horizontal parallax, plus the sun's, 
plus the sun's semi-diameter. 

The angle hEM (Fig. 51,) =EhC'+EC'h. But EhC'=V, the 
moon's horizontal parallax, and EC'^ =8-\-p (Art. 253,) .*. 7iEM 
=P+p+<5, all which are likewise known quantities. 

Hence, by means of these few elements, which are known from 
observation, we ascend to a complete knowledge of all the par- 
ticulars necessary to be known respecting the moon's penumbra. 

255. In the preceding investigations, we have supposed that 
the cone of the earth's shadow is formed by lines drawn from the 
sun, and touching the earth's surface. But the apparent diameter 
of the shadow is found by observation to be somewhat greater than 
would result from this hypothesis. The fact is accounted for by 
supposing that a portion of the solar rays which graze the earth's 
surface are absorbed and extinguished by the lower strata of the 
atmosphere. This amounts to the same thing as though the earth 
were larger than it is, in which case the moon's horizontal parallax 
would be increased ; and accordingly, in order that theory and 
observation may coincide, it is found necessary to increase the 
parallax by gV 

256. In a total eclipse of the moon, its disk is still visible, 
shining with a dull red light. This light cannot be derived di- 
rectly from the sun, since the view of the sun is completely hid- 
den from the moon ; nor by reflexion from the earth, since the 
illuminated side of the earth is wholly turned from the moon ; but 


it is owing to refraction by the earth's atmosphere, by which a few 
scattered rays of the sun are bent round into the earth's shadow 
and conveyed to the moon, sufficient in number to afford the feeble 
light in question. 

257. In calculating an eclipse of the moon, we first learn from 
the tables in what month the sun, at the time of full moon in that 
month, is near the moon's node, or within the lunar ecliptic limit. 
This it must evidently be easy to determine, since the tables ena- 
ble us to find both the longitudes of the nodes, and the longitudes 
of the sun and moon, for every day of the year. Consequently, 
we can find when the sun has nearly the same longitude as one of 
the nodes, and also the precise moment when the longitude of the 
moon is 180 from that of the sun, for this is the time of opposition, 
from which may be derived the time of the middle of the eclipse. 
Having the time of the middle of the eclipse, and the breadth 
of the shadow, (Art. 249,) and knowing, from the tables, the rate 
at which the moon moves per hour faster than the shadow, we can 
find how long it will take her to traverse half the breadth of the 
shadow ; and this time subtracted from the time of the middle 
of the eclipse, will give the beginning, and added to the time of 
the middle will give the end of the eclipse. Or if instead of the 
breadth of the shadow, we employ the breadth of the penumbra 
(Art. 253,) we may find, in the same manner, when the moon 
enters and when she leaves the penumbra. We see, therefore, 
how by having a few things known by observation, such as the 
sun and moon's semi-diameters, and their horizontal parallaxes, 
we rise, by the aid of trigonometry, to the knowledge of various 
particulars respecting the length and breadth of the shadow and 
of the penumbra. These being known, we next have recourse to 
the tables which contain all the necessary particulars respecting 
the motions of the sun and moon, together with equations or cor- 
rections, to be applied for all their irregularities. Hence it is com- 
paratively an easy task to calculate with great accuracy an eclipse 
of the moon. 

258. Let us then see how we may find the exact time of the be- 
ginning, end, duration, and magnitude, of a lunar eclipse. 

150 THE MOON. 

Let NG (Fig 53,) be the ecliptic, and "Nag the moon's orbit, the 
sun being in A* when the moon is in opposition at a ; let N be 
the ascending node, and Aa the moon's latitude at the instant 

Fig. 53. 

of opposition. An hour afterwards the sun will have passed to 
A', and the moon to g, when the difference of longitude of the two 
bodies will be GA'. Then gh is the moon's hourly motion in lati- 
tude, and ah her hourly motion in longitude. As the character 
and form of the eclipse will depend solely upon the distances 
between the centers of the sun and moon, that is, upon the line 
gA', instead of considering the two bodies as both in motion, 
we may suppose the sun at rest in A, and the moon as advancing 
with a motion equal to the difference between its rate and that 
of the sun, a supposition which will simplify the calculation. 
Therefore, draw gd parallel and equal to A'A, join dA, and this 
line being equal to gA', the two bodies will be in the same relative 
situation as if the sun were at A' and the moon at g. Join da and 
produce the line da both ways, cutting the ecliptic in F; then 
da will be the moon's Relative Orbit. Hence aidh AA'=the 
difference of the hourly motions of the sun and moon, that is, the 
moon's relative motion in longitude, and di=ihe moon's hourly 
motion in latitude. 

Draw CD (Fig. 54,) to represent the ecliptic, and let A be the 
place of the sun. As the tables give the computation of the 
moon's latitude at every instant, consequently, we may take from 
them the latitude corresponding to the instant of opposition, and 
to one hour later ; and we may take also the sun's and moon's 
hourly motions in longitude. Take AD, AB, each equal to fhe 
relative motion, and A=the latitude in opposition, Dd=the lati- 

* It will be remarked that the point A really represents the center of the earth's 
shadow ; but as the real motions of the shadow are the same with the assumed motions 
of the sun, the latter are used in conformity with the language of the tables. 


Fig. 54. 

B C 

tude one hour afterwards ; join da and produce the line da both 
ways, and it will represent the moon's relative orbit. Draw B6 
at right angles to CD, and it will be the latitude an hour before 
opposition. At the time of the eclipse, the apparent distance of 
the center of the shadow from the moon is very small ; conse- 
quently, CD, cd, DC?, &c. may be regarded as straight lines. 
During the short interval between the beginning and end of an 
eclipse, the motion of the sun, and consequently that of the cen- 
ter of the shadow, may likewise be regarded as uniform. 

259. The various particulars that enter into the calculation of 
an eclipse are called its Elements ; and as our object is here merely 
to explain the method of calculating an eclipse of the moon, (refer- 
ring to the Supplement for the actual computation,) we may take 
the elements at their mean value. Thus, we will consider cd as 
inclined to CD 5 9', the moon's horizontal parallax as 58', its semi- 
diameter as 16', and that of the earth's shadow as 42'. The line 
Am perpendicular to cd gives the point m for the place of the 
moon at the middle of the eclipse, for this line bisects the chord, 
which represents the path of the moon through the shadow ; and 
mM., perpendicular to CD, gives AM for the time of the middle 
of the eclipse before opposition, the number of minutes before op- 
position being the same part of an hour that AM is of AB.* From 
the center A, with a radius equal to that of the earth's shadow 
(42') describe the semi-circle BLF, and it will represent the pro- 
jection of the shadow traversed by the moon. With a radius 
equal to the semi-diameter of the shadow and that of the moon 

* The situation of the moon when at m is called orbit opposition ; and her situation 
when at a, ecliptic opposition. 

152 THE MOON. 

(= 42'+ 16' =58') and with the center A, mark the two points c and 
f on the relative orbit, and they will be the places of the center 
of the moon at the beginning and end of the eclipse. The per- 
pendiculars cC,/F, give the times AC and AF of the commence- 
ment and the end of the eclipse, and CM, or MF gives half the 
duration. From the centers c and f with a radius equal to the 
semi-diameter of the moon (16') describe circles, and they will 
each touch the shadow, (Euc. 3.12.) indicating the position of the 
moon at the beginning and end of the eclipse. If the same circle 
described from m is wholly within the shadow, the eclipse will be 
total; if it is only partly within the shadow, the eclipse will be 
partial. With the center A, and radius equal to the semi-diame- 
ter of the shadow minus that of the moon (42' 16' =26') mark 
the two points c',f, which will give the places of the center of the 
moon, at the beginning and end of total darkness, and MC', MF' 
will give the corresponding times before and after the middle of 
the eclipse. Their sum will be the duration of total darkness. 

260. If the foregoing projection be accurately made from a scale, 
the required particulars of the eclipse may be ascertained by 
measuring on the same scale, the lines which respectively repre- 
sent them ; and we should thus obtain a near approximation to the 
elements of the eclipse. A more accurate determination of these 
elements may, however, be obtained by actual calculation. The 
general principles of the calculation will be readily understood. 

First, knowing ai, (Fig. 53,) the moon's relative longitude, and 
di, her latitude, we find the angle dai, which is the inclination of 
the moon's relative orbit. But dai=aAm ; and, in the triangle 
aAm, we have the angle at A, and the side A#, being the moon's 
latitude at the time of opposition, which is given by the tables. 
Hence we can find the side Am. In the triangle AmM, (Fig. 54,) 
having the side Am and the angle AmM. (=aAm) we can find AM 
= the arc of relative longitude described by the moon from the 
time of the middle of the eclipse to the time of opposition ; and 
knowing the moon's hourly motion in longitude, we can convert 
AM into time, and this subtracted from the time of opposition 
gives us the time of the middle of the eclipse. 


Secondly, since we know the length of the line Ac* (Fig. 54) 
and can easily find the angle cAC, we can thus obtain the side 
AC ; and AC AM =MC, which arc, converted into time by com- 
paring it with the moon's hourly motion in longitude, gives us, 
when subtracted from the time of the middle of the eclipse, the 
time of the beginning of the eclipse, or when added to that of the 
middle, the time of the end of the eclipse. The sum of the two 
equals the whole duration. 

Thirdly, by a similar method we calculate the value of MC', 
which converted into time, and subtracted from the time of the 
middle of the eclipse, gives the commencement of total darkness, or 
when added gives the end of total darkness. Their sum is the 
duration of total darkness. 

Fourthly, the quantity of the eclipse is determined by supposing 
the diameter of the moon divided into twelve equal parts called 
Digits, and finding- how many such parts lie within the shadow, 
at the time when the centers of the moon and the shadow are 
nearest to each other. Even when the moon lies wholly within 
the shadow, the quantity of the eclipse is still expressed by the num- 
ber of digits contained in that part of the line which joins the cen- 
ter of the 'shadow and the center of the moon, which is intercepted 
between the edge of the shadow and the inner edge of the moon. 

Thus in figure 54, the number of digits eclipsed, equals - - 

T1 7M 

_ o n_ o( mnm) anex p resg ^ on containing only known 


261. The foregoing will serve as an explanation of the generaT 
principles, on which proceeds the calculation of a lunar eclipse. 
The actual methods practiced employ many expedients to facili- 
tate the process, and to insure the greatest possible accuracy, the 
nature of which are explained and exemplified in Mason's Supple- 
ment to this work. 

262. The leading particulars respecting an ECLIPSE or THE 
SUN, are ascertained very nearly like those of a lunar eclipse. The 

* This line is not represented in the figure, but may be easily imagined. 


154 THE MOON. 

shadow of the moon travels over a portion of the earth, as the 
shadow of a small cloud, seen from an eminence in a clear day, 
rides along over hills and plains. Let us imagine ourselves stand- 
ing on the moon ; then we shall see the earth partially eclipsed by 
the shadow of the moon, in the same manner as we now see the 
moon eclipsed by the earth's shadow ; and we might proceed to 
find the length of the shadow, its breadth where it eclipses the 
earth, the breadth of the penumbra, and its duration and quantity, 
in the same way as we have ascertained these particulars for an 
eclipse of the moon. 

But, although the general characters of a solar eclipse might be 
investigated on these principles, so far as respects the earth at 
large, yet as the appearances of the same eclipse of the sun are 
very different at different places on the earth's surface, it is neces- 
sary to calculate its peculiar aspects for each place separately, a 
circumstance which makes the calculation of a solar eclipse much 
more complicated and tedious than of an eclipse of the moon. 
The moon, when she enters the shadow of the earth, is deprived 
of the light of the part immersed, and that part appears black 
alike to all places where the moon is above the horizon. But it is 
not so with a solar eclipse. We do not see this by the shadow 
cast on the earth, as we should do if we stood on the moon, but 
by the interposition of the moon between us and the sun ; and the 
sun may be hidden from one observer while he is in full view of 
another only a few miles distant. Thus, a small insulated cloud 
sailing in a clear sky, will, for a few moments, hide the sun from 
us, and from a certain space near us, while all the region around 
is illuminated. 

263. We have compared the motion of the moon's shadow over 
the surface of the earth to that of a cloud ; but its velocity is in- 
comparably greater. The mean motion of the moon around the 
earth is about 33' per hour ; but 33' of the moon's orbit is 2280 
miles, and the shadow moves of course at the same rate, or 2280 
miles per hour, traversing the entire disk of the earth in less than 
four hours. This is the velocity of the shadow when it passes 
perpendicularly over the earth ; when the direction of the axis of 
the shadow is oblique to the earth's surface, the velocity is increased 


in proportion of radius to the sine of obliquity. Thus the shadows 
of evening have a far more rapid motion than those of noon-day. 
Let us endeavor to form a just conception of the manner in 
which these three bodies, the sun, the earth, and the moon, are 
situated with respect to each other at the time of a solar eclipse. 
First, suppose the conjunction to take place at the node. Then 
the straight line which connects the centers of the sun and the 
earth, also passes through the center of the moon, and coincides 
with the axis of its shadow ; and, since the earth is bisected by 
the plane of the ecliptic, the shadow would traverse the earth in 
the direction of the terrestrial ecliptic, from west to east, passing 
over the middle regions of the earth. Here the diurnal motion of 
the earth being in the same direction with the shadow, but with a 
less velocity, the shadow will appear to move with a speed equal 
only to the difference between the two. Secondly, suppose the 
moon is on the north side of the ecliptic at the time of conjunction, 
and moving towards her descending node, and that the conjunc- 
tion takes place just within the solar ecliptic limit, say 16 from the 
node. The shadow will now not fall in the plane of the ecliptic, 
but a little northward of it, so as just to graze the earth near the 
pole of the ecliptic. The nearer the conjunction comes to the 
node, the further the shadow will fall from the pole of the ecliptic 
towards the equatorial regions. In certain cases, the shadow 
strikes beyond the pole of the earth ; and then its easterly motion 
being opposite to the diurnal motion of the places which it traver- 
ses, consequently its velocity is greatly increased, being equal to 
the sum of both. 

264. After these general considerations, we will now examine 
more particularly the method of investigating the elements of a 
solar eclipse. 

The length of the moon's shadow, is the first object of inquiry. 
The moon, as well as the earth, is at different distances from the 
sun at different times, and hence the length of her shadow varies, 
being always greatest when she is furthest from the sun. Also, 
since her distance from the earth varies, the section of the moon's 
shadow made by the earth, is greater in proportion as the moon is 

156 THE MOON. 

nearer the earth. The greatest eclipses of the sun, therefore, 
happen when the sun is in apogee,* and the moon in perigee. 

265. WJien the moon is at her mean distance from the earth, and 
from the sun, her shadow nearly reaches the earth's surface. 

Let S (Fig. 55,) represent the sun, D the moon, and T the 
earth. Then, the semi-angle of the cone of the moon's shadow, 
DKR, will, as in the case of the earth, (Art. 247,) equal SDR 
DRK, of which SDR is the sun's apparent semi-diameter, as seen 
from the moon, and DRK, is the sun's horizontal parallax at the 
moon. Since, on account of the great distance of the sun, corn- 
Fig. 55. 

pared with that of the moon, the semi-diameter of the sun as seen 
from the moon, must evidently be very nearly the same as 
when seen from the earth, and since on account of the minute- 
ness of the moon's semi-diameter when seen from the sun, the 
sun's horizontal parallax at the moon must be very small, we might, 
without much error, take the siin's apparent semi-diameter from 
the earth, as equal to the semi-angle of the cone of the moon's 
shadow ; but, for the sake of greater accuracy, let us estimate the 
value of the sun's semi-diameter and horizontal parallax at the 

Now, SDR : STR : : ST : SDf : : 400 : 399 ; hence SDR = 

STR=1.0025 STR ; and the sun's mean semi-diameter STR 

being 16.025, hence SDR=1.0025xl6.025=16.065=16' 3".9. 

Again, since parallax is inversely as the distance, the sun's hor- 
izontal parallax at the moon, is on account of her being nearer the 
sun ^ greater than at the earth ; but on account of her inferior 

* The sun is said to be in apogee, when the earth is in aphelion, 
t The apparent magnitude of an object being reciprocally as its distance from the 
eye. See Note, p. 86. 


size it is |f j less than at the earth. Hence, increasing the sun's 
horizontal parallax at the earth by the former fraction, and dimin- 

ishing it by the latter, we have- x ^-x9"=2".5=the sun's 

horizontal parallax at the moon. Therefore, the semi-angle of the 
cone of the moon's shadow, which, as appears above, equals 
SDR DRK, equals 16' 3".9-2".5=16' 1".4, which so nearly 
equals the sun's apparent semi-diameter, as seen from the earth, 
that we may adopt the latter as the value of the semi-angle of the 
shadow. Hence, sin. 16' 1".5 : 1080 (BD) : : Rad. : DK=231690. 
But the mean distance of the moon from the surface of the earth 
is 238545 3956=234589, which exceeds a little the mean length 
of the shadow as above. 

But when the moon is nearest the earth her distance from the 
center of the earth is only 221148 miles; and when the earth is 
furthest from the sun, the sun's apparent semi-diameter is only 
15' 45".5. By employing this number in the foregoing estimate, 
we shall find the length of the shadow 235630 miles; and 
235630221148=14482, the distance which the moon's shadow 
may reach beyond the center of the earth. . 

266. The diameter of the moon's shadow where it traverses the 
earth, is, at its maximum, about 170 miles.* 

In the triangle eTK, the angle at K=15' 45".5 (Art. 265,) the 
side Te=3956, and TK=14482. 

Or, 3956 : 14482 : : sin. 15' 45".5 : sin. 57' 41".5. 

And 57' 41".5+15' 45".5=1 13' 21"=dTe, or the arc de. 

And 2de=2 26' 54"=en. 

Hence 360 : 2.45 (=2 26' 54") : : 24899f : 170 (nearly). 

267. The greatest portion of the earths surface ever covered by 
the moon's penumbra, is about 4393 miles. 

The semi-angle of the penumbra BID=BSD+SBR, of which 
BSD the sun's horizontal parallax at the moon =2". 5, and SBR 
the sun's apparent semi-diameter =16' 3".9, and hence BID is 

* This supposes the conjunction to take place at the node, and the shadow to strike 
the earth perpendicularly to its surface ; where it strikes obliquely, the section may be 
greater than this. 

t The equatorial circumference. 

158 THE MOON. 

known. The moon's apparent semi-diameter BCD =16' 45' .5. 
Therefore GDT is known, as likewise DT and TG. Hence the 
angle GTd may be found, and the arc dG and its double GH, 
which equals the angular breadth of the penumbra. It may be 
converted into miles by stating a proportion as in article 266. 
On making the calculation it will be found to be 4393 miles. 

268. The apparent diameter of the moon is sometimes larger 
than that of the sun, sometimes smaller, and sometimes exactly 
equal to it. Suppose an observer placed on the right line which 
joins the centers of the sun and moon ; if the apparent diameter of 
the moon is greater than that of the sun, the eclipse will be total. 
If the two diameters are equal, the moon's shadow just reaches the 
earth, and the sun is hidden but for a moment from the view of 
spectators situated in the line which the vertex of the shadow de- 
scribes on the surface of the earth. But if, as happens when the 
moon comes to her conjunction in that part of her orbit which is 
towards her apogee, the moon's diameter is less than the sun's, 
then the observer will see a ring of the sun encircle the moon, 
constituting an annular eclipse. (Fig. 55'.) 

Fig. 55'. 

269. Eclipses of the sun are modified by the elevation of the 
moon above the horizon, since its apparent diameter is augmented 


as its altitude is increased, (Art. 217.) This effect, combined with 
that of parallax, may so increase or diminish the apparent distance 
between the centers of the sun and moon, that from this cause 
alone, of two observers at a distance from each other, one might 
see an eclipse which was not visible to the other.* If the hori- 
zontal diameter of the moon differs but little from the apparent 
diameter of the sun, the case might occur where the eclipse would 
be annular over the places where it was observed morning and 
evening, but total where it was observed at mid-day. 

The earth in its diurnal revolution and the moon's shadow both 
move from west to east, but the shadow moves faster than the 
earth ; hence the moon overtakes the sun on its western limb and 
crosses it from west to east. The excess of the apparent diame- 
ter of the moon above that of the sun in a total eclipse is so small, 
that total darkness seldom continues longer than four minutes, and 
can never continue so long as eight minutes. An annular eclipse 
may last 12m. 24s. 

Since the sun's ecliptic limits are more than 17 and the moon's 
less than 12, eclipses of the sun are more frequent than those of 
the moon. Yet lunar eclipses being visible to every part of the 
terrestrial hemisphere opposite to the sun, while those of the sun 
are visible only to the small portion of the hemisphere on which 
the moon's shadow falls, it happens that for any particular place 
on the earth, lunar eclipses are more frequently visible than solar. 
In any year, the number of eclipses of both luminaries cannot be 
ess than two nor more than seven : the most usual number is four, 
and it is very rare to have more than six. A total eclipse of the 
moon frequently happens at the next full moon after an eclipse of 
the sun. For since, in an eclipse of the sun, the sun is at or near 
one of the moon's nodes, the earth's shadow must be at or near 
the other node, and may not have passed so far from the node as 
the lunar ecliptic limits, before the moon overtakes it. 

270. It has been observed already, that were the spectator on 
the moon instead of on the earth, he would see the earth eclipsed 
by the moon, and the calculation of the eclipse would be very sim- 
ilar to that of a lunar eclipse ; but to an observer on the earth the 

* Biot, Ast. Phys. p. 401. 

160 THE MOON. 

eclipse does not of course begin when the earth first enters the 
moon's shadow, and it is necessary to determine not only what 
portion of the earth's surface will be covered by the moon's sha- 
dow, but likewise the path described by its center relative to va- 
rious places on the surface of the earth. This is known when the 
latitude and longitude of the center of the shadow on the earth, is 
determined for each instant. The latitude and longitude of the 
moon are found on the supposition that the spectator views it from 
the center of the earth, whereas his position on the surface changes, 
in consequence of parallax, both the latitude and longitude, and 
the amount of these changes must be accurately estimated, before 
the appearance of the eclipse at any particular place can be fully 

The details of the method of calculating a solar eclipse cannot 
be understood in any way so well, as by actually performing the 
process according to a given example. For such details therefore 
the reader is referred to the Supplement. 

271. In total eclipses of the sun, there has sometimes been ob- 
served a remarkable radiation of light from the margin of the sun. 
This has been ascribed to an illumination of the solar atmosphere, 
but it is with more probability owing to the zodiacal light (Art. 
152,) which at that time is projected around the sun, and which is 
of such dimensions as to extend far beyond the solar orb.* 

A total eclipse of the sun is one of the most sublime and impres- 
sive phenomena of nature. Among barbarous tribes it is ever con- 
templated with fear and astonishment, while among cultivated na- 
tions it is recognized, from the exactness with which the time of 
occurrence and the various appearances answer to the prediction, 
as affording one of the proudest triumphs of astronomy. By 
astronomers themselves it is of course viewed with the highest 
interest, not only as verifying their calculations, but as contribu- 
ting to establish beyond all doubt the certainty of those grand 
laws, the truth of which is involved in the result. During the 
eclipse of June, 1806, which was one of the most remarkable on 

* See an excellent description and delineation of this appearance as it was exhibited 
in the eclipse of 1806, in the Transactions of the Albany Institute, by the late Chan, 
eellor De Witt, 


record, the time of total darkness, as seen by the author of this 
work, was about mid-day. The sky was entirely cloudless, but 
as the period of total obscuration approached, a gloom pervaded 
all nature. When the sun was wholly lost sight of, planets and 
stars came into view ; a fearful pall hung upon the sky, unlike 
both to night and to twilight ; and, the temperature of the air rap- 
idly declining, a sudden chill came over the earth. Even the ani- 
mal tribes exhibited tokens of fear and agitation. 

From 1831 to 1838 was a period remarkable for great eclipses 
of the sun, in which time there were no less than five of the most 
remarkable character. The next total eclipse of the sun, visible 
in the United States, will occur on the 7th of August, 1869. 



272. As eclipses of the sun afford one of the most approved 
methods of finding the longitudes of places, our attention is natu- 
rally turned next towards that subject. 

The ancients studied astronomy in order that they might read 
their destinies in the stars : the moderns, that they may securely 
navigate the ocean. A large portion of the refined labors of 
modern astronomy, has been directed towards perfecting the as- 
tronomical tables with the view of finding the longitude at sea, 
an object manifestly worthy of the highest efforts of science, con- 
sidering the vast amount of property and of human life involved 
in navigation. 

273. The difference of longitude between two places may be found 
by any method, by which we can ascertain the difference of their local 
times, at the same instant of absolute time. 

As the earth turns on its axis from west to east, any place that 
lies eastward of another will come sooner under the sun, or will 


162 THE MOON. 

have the sun earlier on the meridian, and consequently, in respect 
to the hour of the day, will be in advance of the other at the 
rate of one hour for every 15, or four minutes of time for each 
degree. Thus, to a place 15 east of Greenwich, it is 1 o'clock, 
P. M. when it is noon at Greenwich; and to a place 15 west of 
that meridian, it is 11 o'clock, A. M. at the same instant. Hence, 
the difference of time at any two places, indicates their difference 
of longitude. 

274. The easiest method of finding the longitude is by means 
of an accurate time piece, or chronometer. Let us set out from 
London with a chronometer accurately adjusted to Greenwich 
time, and travel eastward to a certain place, where the time is 
accurately kept, or may be ascertained by observation. We find, 
for example, that it is 1 o'clock by our chronometer, when it is 
2 o'clock and 30 minutes at the place of observation. Hence, 
the longitude is 15x1.5=22^ E. Had we travelled westward 
until our chronometer was an hour and a half in advance of the 
time at the place of observation, (that is, so much later in the 
day,) our longitude would have been 22| W. But it would not 
be necessary to repair to London in order to set our chronometer 
to Greenwich time. This might be done at any observatory, or 
any place whose longitude had been accurately determined. For 
example, the time at New York is 4h. 56m. 4 8 .5 behind that of 
Greenwich. If, therefore, we set our chronometer so much be- 
fore the true time at New York, it will indicate the time at Green- 
wich. Moreover, on arriving at different places, any where on 
the earth, whose longitude is accurately known, we may learn 
whether our chronometer keeps accurate time or not, and if not, 
the amount of its error. Chronometers have been constructed of 
such an astonishing degree of accuracy, as to deviate but a few 
seconds in a voyage from London to Baffin's Bay and back, during 
an absence of several years. But chronometers which are suffi- 
ciently accurate to be depended on for long voyages, are too ex- 
pensive for general use, and the means of verifying their accuracy 
are not sufficiently easy. Moreover, chronometers by being trans- 
ported from one place to another, change their daily rate, or de- 
part from that mean rate which they preserve at a fixed station. 


A chronometer, therefore, cannot be relied on for determining the 
longitudes of places where the greatest degree of accuracy is re- 
quired, especially where the instrument is conveyed over land, 
although the uncertainty attendant on one instrument may be 
nearly obviated by employing several and taking their mean 

275. Eclipses of the sun and moon are sometimes used for de- 
termining the longitude. The exact instant of immersion or of 
emersion, or any other definite moment of the eclipse which pre- 
sents itself to two distant observers, affords the means of com- 
paring their difference of time, and hence of determining their 
difference of longitude. Since the entrance of the moon into 
the earth's shadow, in a lunar eclipse, is seen at the same instant 
of absolute time at all places where the eclipse is visible, (Art. 
262,) this observation would be a very suitable one for finding 
the longitude were it not that, on account of the increasing dark- 
ness of the penumbra near the boundaries of the shadow, it is 
difficult to determine the precise instant when the moon enters the 
shadow. By taking observations on the immersions of known 
spots on the lunar disk, a mean result may be obtained which will 
give the longitude with tolerable accuracy. In an eclipse of the 
sun, the instants of immersion and emersion may be observed with 
greater accuracy, although, since these do not take place at the 
same instant of absolute time, the calculation of the longitude from 
observations on a solar eclipse are complicated and laborious. 

A method very similar to the foregoing, by observations on 
eclipses of Jupiter's satellites, and on occultations of stars, will 
be mentioned hereafter. 

276. The Lunar method of finding the longitude, at sea, is in 
many respects preferable to every other. It consists in measuring 
(with a sextant) the angular distance between the moon and the 
sun, or between the moon and a star, and then turning to the Nau- 
tical Almanac,f and finding what time it was at Greenwich when 

* Woodhouse, p. 838. 

t The Nautical Almanac is a book published annually by the British Board of 
Longitude, containing various tables and astronomical information for the use oi 

164 THE MOON. 

that distance was the same. The moon moves so rapidly, that this 
distance will not be the same except at very nearly the same in- 
stant of absolute time. For example, at 9 o'clock, A. M., at a cer- 
tain place, we find the angular distance of the moon and the sun to 
be 72 ; and on looking into the Nautical Almanac, we find that 
at the time when this distance was the same for the meridian of 
Greenwich was 1 o'clock, P. M. ; hence we infer that the longi- 
tude of the place is four hours, or 60 west. 

The Nautical Almanac contains the true angular distance of 
the moon from the sun, from the four large planets, (Venus, Mars, 
Jupiter, and Saturn,) and from nine bright fixed stars, for the be- 
ginning of every third hour of mean time for the meridian of 
Greenwich ; and the mean time corresponding to any intermediate 
hour, may be found by proportional parts.* 

277. It would be a very simple operation to determine the lon- 
gitude by Lunar Distances, if the process as described in the 
preceding article were all that is necessary ; but the various cir- 
cumstances of parallax, refraction, and dip of the horizon, would 
differ more or less at the two places, even were the bodies whose 
distances were taken in view from both, which is not necessarily 
the case. The observations, therefore, require to be reduced to 
the center of the earth, being cleared of the effects of parallax and 
refraction. Hence, three observers are necessary in order to take 
a lunar distance in the most exact manner, viz. two to measure 
the altitudes of the two bodies respectively, at the same time that 
the third takes the angular distance between them. The altitudes 
of the two luminaries at the time of observation must be known, 
in order to estimate the effects of parallax and refraction. 

278. Although the lunar method of finding the longitude at 
sea has many advantages over the other methods in use, yet it 

navigators. The American Almanac also contains a variety of astronomical informa- 
tion, peculiarly interesting to the people of the United States, in connexion with a 
vast amount of statistical matter. It is well deserving a place in the library of the 

* See Bowditch's Navigator, Tenth Ed. p. 226. 

TIDES. 165 

has also its disadvantages. One is, the great exactness requisite 
in observing the distance of the moon from the sun or star, as a 
small error in the distance makes a considerable error in the longi- 
tude. The moon moves at the rate of about a degree in two 
hours, or one minute of space in two minutes of time. There- 
fore, if we make an error of one minute in observing the distance, 
we make an error of two minutes in time, or 30 miles of longitude 
at the equator. A single observation with the best sextants, may 
be liable to an error of more than half a minute ; but the accuracy 
of the result may be much increased by a mean of several obser- 
vations taken to the east and west of the moon. The imperfection 
of lunar tables was until recently considered as an objection to this 
method. Until within a few years, the best lunar tables were 
frequently erroneous to the amount of one minute, occasioning an 
error of 30 miles. The error of the best tables now in use will 
rarely exceed 7 or 8 seconds.* 


279. The tides are an alternate rising and falling of the waters 
of the ocean, at regular intervals. They have a maximum and a 
minimum twice a day, twice a month, and twice a year. Of the 
daily tide, the maximum is called High tide, and the minimum 
Low tide. The maximum for the month is called Spring tide, and 
the minimum Neap tide. The rising of the tide is called Flood 
and its falling Ebb tide. 

Similar tides, whether high or low, occur on opposite sides of 
the earth at once. Thus at the same time it is high tide at any 
given place, it is also high tide on the inferior meridian, and the 
same is true of the low tides. 

The interval between two successive high tides is 12h. 25m. ; 
or, if the same tide be considered as returning to the meridian, 
after having gone around the globe, its return is about 50 minutes 
later than it occurred on the preceding day. In this respect, as 
well as in various others, it corresponds very nearly to the motions 
of the moon. 

* Brinkley's Elements of Astronomy, p. 241 

166 THE MOON. 

The average height for the whole globe is about 2 feet; 01, 
if the earth were covered uniformly with a stratum of water, the 
difference between the two diameters of the oval would be 5 feet, 
or more exactly 5 feet and 8 inches ; but its natural height at 
various places is very various, sometimes rising to 60 or 70 feet, 
and sometimes being scarcely perceptible. At the same place 
also the phenomena of the tides are very different at different 

Inland lakes and seas, even those of the largest class, as Lake 
Superior, or the Caspian, have no perceptible tide. 

280. Tides are caused by the unequal attraction of the sun and 
moon upon different parts of the earth. 

Suppose the projectile force by which the earth is carried for- 
ward in her orbit, to be suspended, and the earth to fall towards 
one of these bodies, the moon, for example, in consequence of 
their mutual attraction. Then, if all parts of the earth fell 
equally towards the moon, no derangement of its different parts 
would result, any more than of the particles of a drop of water 
in its descent to the ground. But if one part fell faster than an- 
other, the different portions would evidently be separated from 
each other. Now this is precisely what takes place with respect 
to the earth in its fall towards the moon. The portions of the 
earth in the hemisphere next to the moon, on account of being 
nearer to the center of attraction, fall faster than those in the op- 
posite hemisphere, and consequently leave them behind. The 
solid earth, on account of its cohesion, cannot obey this impulse, 
since all its different portions constitute one mass, which is acted 
on in the same manner as though it were all collected in the cen- 
ter ; but the waters on the surface, moving freely under this im- 
pulse, endeavor to desert the solid mass and fall towards the 
moon. For a similar reason the waters in the opposite hemisphere 
falling less towards the moon than the solid earth, are left behind, 
or appear to rise from the center of the earth. 

281. Let DEFG (Fig. 56,) represent the globe ; and, for the sake 
of illustrating the principle, we will suppose the waters entirely to 
cover the globe at a uniform depth. Let defg represent the solid 



globe, and the circular ring exterior to Fig. 56. 

it, the covering of waters. Let C be 
the center of gravity of the solid mass, 
A that of the hemisphere next to the 
moon, and B that of the remoter hemi- 
sphere. Now the force of attraction 
exerted by the moon, acts in the same 
manner as though the solid mass were 
all concentrated in C, and the waters 
of each hemisphere at A and B respec- 
tively ; and (the moon being supposed above E) it is evident that 
A will tend to leave C, and C to leave B behind. The same must 
evidently be true of the respective portions of matter, of which 
these points are the centers of gravity. The waters of the globe 
will thus be reduced to an oval shape, being elongated in the direc- 
tion of that meridian which is under the moon, and flattened in 
the intermediate parts, and most of all at points ninety degrees dis- 
tant from that meridian. 

Were it not, therefore, for impediments which prevent the force 
from producing its full effects, we might expect to see the great 
tide-wave, as the elevated crest is called, always directly beneath 
the moon, attending it regularly around the globe. But the in- 
ertia of the waters prevents their instantly obeying the moon's 
attraction, and the friction of the waters on the bottom of the 
ocean, still further retards its progress. It is not therefore until 
several hours (differing at different places) after the moon has 
passed the meridian of a place, that it is high tide at that place. 

282. The sun has a similar action to the moon, but only one 
third as great. On account of the great mass of the sun com- 
pared with that of the moon, we might suppose that his action 
in raising the tides would be greater than the moon's ; but the 
nearness of the moon to the earth more than compensates for 
the sun's greater quantity of matter. Let us, however, form a 
just conception of the advantage which the moon derives from her 
proximity. It is not that her actual amount of attraction is thus 
rendered greater than that of the sun ; but it is that her attraction 
for the different parts of the earth is very unequal, while that of 

168 THE MOON. 

the sun is nearly uniform. It is the inequality of this action, and 
not the absolute force, that produces the tides. The diameter of 
the earth is ^ of the distance of the moon, while it is less than 
roioT of the distance of the sun. 

283. Having now learned the general cause of the tides, we 
will next attend to the explanation of particular phenomena. 

The Spring tides, or those which rise to an unusual height 
twice a month, are produced by the sun and moon's acting to- 
gether; and the Neap tides, or those which are unusually low 
twice a month, are produced by the sun and moon's acting in 
opposition to each other. The Spring tides occur at the syzygies ; 
the Neap tides at the quadratures. At the time of new moon, 
the sun and moon both being on the same side of the earth, and 
acting upon it in the same line, their actions conspire, and the 
sun may be considered as adding so much to the force of the 
moon. We have already explained how the moon contributes to 
raise a tide on the opposite side of the earth. But the sun as well 
as the moon raises its own tide-wave, which, at new moon, coin- 
cides with the lunar tide-wave. At full moon, also, the two lumina- 
ries conspire in the same way to raise the tide ; for we must recol- 
lect that each body contributes to raise the tide 6n the opposite 
side of the earth as well as on the side nearest to it. At both the 
conjunctions and oppositions, therefore, that is, at the syzygies, 
we have unusually high tides. But here also the maximum effect 
is not at the moment of the syzygies, but 36 hours afterwards. 

At the quadratures, the solar wave is lowest where the lunar 
wave is highest ; hence the low tide produced by the sun is sub- 
tracted from high water and produces the Neap tides. Moreover, 
at the quadratures the solar wave is highest where the lunar wave 
is lowest, and hence is to be added to the height of low water at 
the time of Neap tides. Hence the difference between high and 
low water is only about half as great at Neap tide as at Spring tide. 

284. The power of the moon or of the sun to raise the tide is 
found by the doctrine of universal gravitation to be inversely as 
the cube of the distance* The variations of distance in the sun are 

* La Place, Syst, du Monde, 1. iv, c. x. 



not great enough to influence the tides very materially, but the 
variations in the moon's distances have a striking effect. The 
tides which happen when the moon is in perigee, are considerably 
greater than when she is in apogee ; and if she happens to be in 
perigee at the time of the syzygies, the spring tide is unusually 
high. When this happens at the equinoxes, the highest tides of 
the year are produced. 

285. The declinations of the sun and moon have a considerable 
influence on the height of the tide. When the moon, for example, 
has no declination, or is in the equator, as in figure 57,* the rota- 
tion of the earth on its axis NS will make the two tides exactly 
equal on opposite sides of the earth. Thus a place which is car- 
ried through the parallel TT' will have the height of one tide T2 
and the other tide T'3. The tides are in this case greatest at the 
equator, and diminish gradually to the poles, where it will be low 
water during the whole day. When the moon is on the north side 
of the equator, as in figure 58, at her greatest northern declination, 
Fig. 57. Fig. 58. 

a place describing the parallel TT' will have T'3 for the height of 
the tide when the moon is on the superior meridian, and T2 for 
the height when the moon is on the inferior meridian. Therefore, 
all places north of the equator will have the highest tide when the 
moon is above the horizon, and the lowest when she is below it ; 
the difference of the tides diminishing towards the equator, where 

* Diagrams like these are apt to mislead the learner, by exhibiting the protuberance 
occasioned by the tides as much greater than the reality. We must recollect that it 
amounts, at the highest, to only a very few feet in eight thousand miles. Were the 
diagram, therefore, drawn in just proportions, the alterations of figure produced by the 
lides would be wholly insensible. 


170 THE MOON. 

they are equal. In like manner, places south of the equator have 
the highest tides when the moon is below the horizon, and the 
lowest when she is above it. When the moon is at her greatest 
declination, the highest tides will take place towards the tropics. 
The circumstances are all reversed when the moon is south of the 

280. The motion of the tide- wave, it should be remarked, is not 
a progressive motion, but a mere undulation, and is to be carefully 
distinguished from the currents to which it gives rise. If the 
ocean completely covered the earth, the sun and moon being in the 
equator, the tide-wave would travel at the same rate as the earth 
on its axis. Indeed, the correct way of conceiving of the tide- 
wave, is to consider the moon at rest, and the earth in its rotation 
from west to east as bringing successive portions of water under 
the moon, which portions being elevated successively at the same 
rate as the earth revolves on its axis, have a relative motion west- 
ward in the same degree. 

287. The tides of rivers, narrow bays, and shores far from the 
main body of the ocean, are not produced in those places by the 
direct action of the sun and moon, but are subordinate waves 
propagated from the great tide-wave. 

Lines drawn through all the adjacent parts of any tract of wa- 
ter, which have high water at the same time, are called cotidal 
lines.^ We may, for instance, draw a line through all places in 
the Atlantic Ocean which have high tide on a given day at 1 o'clock, 
and another through all places which have high tide at 2 o'clock. 
The cotidal line for any hour may be considered as representing 
the summit or ridge of the tide- wave at that time ; and could the 
spectator, detached from the earth, perceive the summit of the 
wave, he would see it travelling round the earth in the open ocean 
once in twenty four hours, followed by another twelve hours dis- 
tant, and both sending branches into rivers, bays, and other open- 
ings into the main land. These latter are called Derivative tides, 

* Edin. Encyc. Art. Astronomy, p. 623. 

t Whewell, Phil. Transaction for 1833, p. 148. 

TIDES. 171 

while those raised directly by the action of the sun and moon, are 
called Primitive tides. 

288. The velocity with which the wave moves will depend on 
various circumstances, but principally on the depth, and probably 
on the regularity of the channel. If the depth be nearly uniform, 
the cotidal lines will be nearly straight and parallel. But if some 
parts of the channel are deep while others are shallow, the tide 
will be detained by the greater friction of the shallow places, and 
the cotidal lines will be irregular. The direction also of the de- 
rivative tide, may be totally different from that of the primitive. 
Thus, (Fig. 59,) if the great tide- Fig. 59. 

wave, moving from east to west, 
be represented by the lines 1, 2, 
3, 4, the derivative tide which is 
propagated up a river or bay, 
will be represented by the cotidal 
lines 3, 4, 5, 6, 7. Advancing 
faster in the channel than next 
the banks, the tides will lag be- 
hind towards the shores, and the 
cotidal lines will take the form 
of curves as represented in the 

289. On account of the retarding influence of shoals, and an 
uneven, indented coast, the tide-wave travels more slowly along 
the shores of an island than in the neighboring sea, assuming con- 
vex figures at a little distance from the island and on opposite 
sides of it. These convex lines sometimes meet and, become 
blended in such a manner as to create singular anomalies in a sea 
much broken by islands, as well as on coasts indented with numer- 
ous tfays arid rivers.* Peculiar phenomena are also produced, 
when the tide flows in at opposite extremities of a reef or island, 
as into the two opposite ends of Long Island Sound. In certain 

* See an excellent representation and description of these different phenomena by 
Professor Whewell, Phil. Trans. 1833, p. 153. 

172 THE MOON. 

cases a tide-wave is forced into a narrow arm of the sea, and 
produces very remarkable tides. The tides of the Bay of Fundy 
(the highest in the world) sometimes rise to the height of 60 or 70 
feet ; and the tides of the river Severn, near Bristol in England, 
rise to the height of 40 feet. 

290. The Unit of Altitude of any place, is the height of the 
maximum tide after the syzygies, (Art. 283,) being usually about 
36 hours after the new or full moon. But as the amount of this 
tide would be affected by the distance of the sun and moon from 
the earth, (Art. 284,) and by their declinations, (Art. 285,) these 
distances are faken at their mean value, and the luminaries are 
supposed to be in the equator ; the observations being so reduced 
as to conform to these circumstances. The unit of altitude can be 
ascertained by observation only. The actual rise of the tide de- 
pends much on the strength and direction of the wind. When 
high winds conspire with a high flood tide, as is frequently the 
case near the equinoxes, the tide rises to a very unusual height. 
We subjoin from the American Almanac a few examples of the 
unit of altitude for different places. 


Cumberland, head of the Bay of Fundy, 71 

Boston, . . . . 11J 

New Haven, .... 8 
New York, .... 5 
Charleston, S. C., ... 6 

291. The Establishment of any port is the mean interval between 
noon and the time of high water, on the day of new or full moon. 
As the interval for any given place is always nearly the same, it 
becomes a criterion of the retardation of the tides at that place. 
On account of the importance to navigation of a correct know- 
ledge of the tides, the British Board of Admiralty, at the sugges- 
tion of the Royal Society, recently issued orders to their agents 
in various important naval stations, to have accurate observations 
made on the tides, with the view of ascertaining the establishment 
and various other particulars respecting each station;* and the 

* Lubbock, Report on the Tides, 1833. 

TIDES. 173 

government of the United States is prosecuting similar investiga- 
tions respecting our own ports. 

292. According to Professor Whewell,* the tides on the coast 
of North America are derived from the great tide-wave of the 
South Atlantic, which runs steadily northward along the coast to 
the mouth of the Bay of Fundy, where it meets the northern tide 
wave flowing in the opposite direction. Hence he accounts for 
the high tides of the Bay of Fundy. 

293. The largest lakes and inland seas have no perceptible 
tides. This is asserted by all writers respecting the Caspian and 
Euxine, and the same is found to be true of the largest of the 
North American lakes, Lake Superior. f 

Although these several tracts of water appear large when taken 
by themselves, yet they occupy but small portions of the surface 
of the globe, as will appear evident from the delineation of them 
on an artificial globe. Now we must recollect that the primitive 
tides are produced by the unequal action of the sun and moon 
upon the different parts of the earth ; and that it is only at points 
whose distance from each other bears a considerable ratio to the 
whole distance of the sun or the moon, that the inequality of ac- 
tion becomes manifest. The space required is larger than either 
of these tracts of water. It is obvious also that they have no op- 
portunity to be subject to a derivative tide. 

294. To apply the theory of universal gravitation to all the va- 
rying circumstances that influence the tides, becomes a matter of 
such intricacy, that La Place pronounces " the problem of the 
tides" the most difficult problem of celestial mechanics. 

295. The Atmosphere that envelops the earth, must evidently be 
subject to tfre action of the same forces as the covering of waters, 
and hence we might expect a rise and fall of the barometer, indi- 
cating an atmospheric tide corresponding to the tide of the ocean. 

* Phil. Trans. 1833, p. 172. 

t See Experiments of Gov. Cass, Am. Jour. Science. 


La Place has calculated the amount of this aerial tide. It is too 
inconsiderable to be detected by changes in the barometer, unless 
by the most refined observations. Hence it is concluded, that the 
fluctuations produced by this cause are too slight to affect me- 
teorological phenomena in any appreciable degree.* 



296. THE name planet signifies a wanderer^ and is applied to 
this class of bodies because they shift their positions in the heav 
ens, whereas the fixed stars constantly maintain the same places 
with respect to each other. The planets known from a high an- 
tiquity, are Mercury, Venus, Earth, Mars, Jupiter, and Saturn. 
To these, in 1781, was added Uranus, J (or Herschel, as it is some- 
times called from the name of its discoverer,) and, as late as the 
commencement of the present century, four more were added, 
namely, Ceres, Pallas, Juno, and Vesta. These bodies are desig- 
nated by the following characters : 

1. Mercury S 7. Ceres ? 

2. Venus 9 8. Pallas $ 

3. Earth 9. Jupiter U 

4. Mars <? 10. Saturn ^ 

5. Vesta fi 11. Uranus ^ 

6. Juno $ 

The foregoing are called the primary planets. Several of these 
have one or more attendants, or satellites, which revolve around 
them, as they revolve around the sun. The earth has one satel- 
lite, namely, the moon ; Jupiter has four ; Saturn, seven ; and Ura- 

* Bowditch's La Place, II. 797. 
t From the Greek, 
t From 


nus, six. These bodies also are planets, but in distinction from the 
others they are called secondary planets. Hence, the whole num 
ber of planets are 29, viz. 11 primary, and 18 secondary planets.* 

297. With the exception of the four new planets, these bodies 
have their orbits very nearly in the same plane, and are never seen 
far from the ecliptic. Mercury, whose orbit is most inclined of 
all, never departs further from the ecliptic than about 7, while 
most of the other planets pursue very nearly the same path with 
the earth, in their annual revolution around the sun. The new 
planets, however, make wider excursions from the plane of the 
ecliptic, amounting, in the case of Pallas, to 34. 

298. Mercury and Yenus are called inferior planets, because 
they have their orbits nearer to the sun than that of the earth ; 
while all the others, being more distant from the sun than the 
earth, are called superior planets. The planets present great di- 
versities among themselves in respect to distance from the sun, 
magnitude, time of revolution, and density. They differ also in 
regard to satellites, of which, as we have seen, three have respec- 
tively four, six, and seven, while more than half have none at all. 
It will aid the memory, and render our view of the planetary sys- 
tem more clear and comprehensive, if we classify, as far as possi- 
ble, the various particulars comprehended under the foregoing 


1. Mercury, 



2. Venus, 



3. Earth, 



4. Mars, 



5. Vesta, 



6. Juno, 



7. Ceres, 

> 261,000,000 


8. Pallas, 



* See Article V. of the Addenda. 

t The distance in miles, as expressed in the first column, in round numbers, is to be 
treasured up in the memory, while the second column expresses the relative distances, 
that of the earth being 1, from which a more exact determination may be made, when 
required, the earth's distance being taken at 94,885,491. (Daily.) 


9. Jupiter, 485,000,000 5.2027760 

10. Saturn, 890,000,000 9.5387861 

11. Uranus, 1800,000,000 19.1823900 

The dimensions of the planetary system are seen from this 
table to be vast, comprehending a circular space thirty-six hun- 
dred millions of miles in diameter. A railway car, travelling con- 
stantly at the rate of 20 miles an hour, would require more than 
20,000 years to cross the orbit of Uranus. 

It may aid the memory to remark, that in regard to the planets 
nearest the sun, the distances increase in an arithmetical ratio, 
while those most remote increase in a geometrical ratio. Thus, 
if we add 30 to the distance of Mercury, it gives us nearly that of 
Venus ; 30 more gives that of the Earth ; while Saturn is nearly 
twice the distance of Jupiter, and Uranus twice the distance of 
Saturn. Between the orbits of Mars and Jupiter, a great chasm 
appeared, which broke the continuity of the series ; but the dis- 
covery of the new planets has filled the void. A more exact law 
of the series was discovered a few years since by Mr. Bode of 
Berlin. It is as follows : if we represent the distance of Mercury 
by 4, and increase each term by the product of 3 into a certain 
power of 2, we shall obtain the distances of each of the planets in 
succession. Thus, 

Mercury, ... 4 =4 

Venus, .... 4+3.2 = 7 

Earth, .... 4+3.2 1 = 10 

Mars, .... 4+S.2 2 = 16 

Ceres, .... 4+3.2 3 = 28 

Jupiter, . . . . ^ 4+3.2 4 = 52 

Saturn, . . . . * 4+3.2 5 =100 

Uranus, .... 4+3.2 6 =196 

For example, by this law, the distances of the Earth and Jupi- 
ter are to each other as 10 to 52. Their actual distances as given 
in the table (Art. 299,) are as 1 to 5.202776 ; but 1 : 5.202776 : : 
10 : 52 nearly. 

The mean distances of the planets from the sun, may also be de- 
termined by means of Kepler's law, that the squares of the period- 


ical times are as the cubes of the distances, (Art. 192.) Thus the 
earth's distance being previously ascertained by means of the 
sun's horizontal parallax, (Art. 87,) and the period of any other 
planet as Jupiter, being learned from observation, we say as 
365T256 2 : 4332.585** : : I 3 : 5.202 3 . But 5.202 is the number, 
which, according to the table, (Art. 299,) expresses the distance of 
Jupiter from the sun. 


Diam. in Miles. Mean apparent Diam. Volume. 

Mercury, . . . 3140 6".9 ^ 

Venus, .... 7700 16".9 T V 

Earth, .... 7912 1 

Mars, .... 4200 6".3 | 

Ceres, .... 160 0".5 

Jupiter, .... 89000 36".7 1281 

Saturn .... 79000 16".2 995 

Uranus .... 35000 4".0 80 

We remark here a great diversity in regard to magnitude, a 
diversity which does not appear to be subject to any definite 
law. While Venus, an inferior planet, is T 9 y as large as the earth, 
Mars, a superior planet, is only |, while Jupiter is 1281 times as 
large. Although several of the planets, when nearest to us, appear 
brilliant and large when compared with the fixed stars, yet the 
angle which they subtend is very small, that of Venus, the great- 
est of all, never exceeding about 1', or more exactly 61".2, and 
that of Jupiter being when greatest only about f of a minute. 

The distance of one of the near planets, as Venus or Mars, may 
be determined from its parallax ; and the distance being known, 
its real diameter can be estimated from its apparent diameter, in 
the same manner as we estimate the diameter of the sun. (Art. 

* This is the number of days in one revolution of Jupiter. 



Revolution in its orbit Mean daily motion. 

Mercury 3 months, or 88 days, 4 5' 32".6 
Venus, 7i " " 224 " 1 36' 7 // .8 

Earth, 1 year, " 365 " 59' 8".3 

Mars, 2 " " 687 " 31' 26".7 

Ceres, 4 " " 1681 " 12' 50".9 

Jupiter, 12 " " 4332 " 4' 59".3 

Saturn, 29 " " 10759 " 2' 0".6 

Uranus, 84 " " 30686 " 0' 42".4 

From this view, it appears that the planets nearest the sun move 
most rapidly. Thus Mercury performs nearly 350 revolutions 
while Uranus performs one. This is evidently not owing merely 
to the greater dimensions of the orbit of Uranus, for the length of 
its orbit is not 50 times that of the orbit of Mercury, while the 
time employed in describing it is 350 times that of Mercury. In- 
deed this ought to follow from Kepler's law that the squares of 
the periodical times are as the cubes of the distances, from which 
it is manifest that the times of revolution increase faster than the 
dimensions of the orbit. Accordingly, the apparent progress of 
the most distant planets is exceedingly slow, the daily rate of Ura- 
nus being only 42".4 per day ; so that for weeks and months, and 
even years, this planet but slightly changes its place among the 


302. The inferior planets, Mercury and Venus, having their or- 
bits so far within that of the earth, appear to us as attendants upon 
the sun. Mercury never appears further from the sun than 29 
(28 48') and seldom so far ; and Venus never more than about 
47 (47 12'). Both planets, therefore, appear either in the west 
soon after sunset, or in the east a little before sunrise. In high 
latitudes, where the twilight is prolonged, Mercury can seldom be 
seen with the naked eye, and then only at the periods of its great- 
est elongation.* The reason of this will readily appear from the 
following diagram. 

* Copernicus is said to have lamented on his death-bed that he had never been able 
to obtain a sight of Mercury, and Delambre saw it but twice. 



Let S (Fig. 60,) represent the sun, ADB the orbit of Mercury, 
and E the place of the Earth. Each of the planets is seen at its 
greatest elongation, when a line, EA or EB in the figure, is a tan- 
gent to its orbit. Then the sun being at S' in the heavens, the 
planet will be seen at A' and B', when at its greatest elongations, 
and will appear no further from the sun than the arc S'A' or S'B' 

Fig. 60. 

303. A planet is said to be in conjunction with the sun, when it 
is seen in the same part of the heavens with the sun, or when it 
has the same longitude. Mercury and Venus have each two con- 
junctions, the inferior and the superior. The inferior conjunction 
is its position when in conjunction on the same side of the sun 
with the earth, as at C in the figure : the superior conjunction is its 
position when on the side of the sun most distant from the earth, 
as at D. 

304. The period occupied by a planet between two successive 
conjunctions with the earth, is called its si/nodical revolution. 
Both the planet and the earth being in motion, the time of the 
synodical revolution exceeds that of the sidereal revolution of 
Mercury or Venus ; for when the planet comes round to the place 
where it before overtook the earth, it does not find the earth at 
that point, byt far in advance of it. Thus, let Mercury come into 


inferior conjunction with the earth at C, (Fig. 60.) In about 88 
days, the planet will come round to the same point again; but 
meanwhile the earth has moved forward through the arc EE', and 
will continue to move while the planet is moving more rapidly to 
overtake her, the case being analogous to that of the hour and 
second hand of a clock. 

Having the sidereal period of a planet, (which may always be 
accurately determined by observation,) we may ascertain its sy- 
nodical period as follows. Let T denote the sidereal period of 
the earth, and T 7 that of the planet, Since, in the time T the 


earth describes a complete revolution, T : T' : : 1 : -^ = the part 

of the circumference described by the earth in the time T f . But 
during the same time the planet describes a whole circumference. 


Therefore, 1 -^ is what the planet gains on the earth in one 

revolution. In order to a new conjunction the planet must gain 
an entire circumference ; therefore, denoting the synodical period 
by S, the gain in one revolution will be to the time in which it 
is acquired, as a whole circumference is to the time in which that 
is gained, which is the synodical period. That is, 

TV r r r r/ 

1 T' 1 -S 

* ryi * **, 'T' 'TV* 

From this formula we may find the synodical revolution of Mer- 
cury or Venus, by substituting the numbers denoted by the letters. 

Thus, 365 ' 256X87 ' 969 =:115.877, which is the synodical period 


of Mercury. 

By a similar computation, the synodical revolution of Venus 
will be found to be about 584 days. 

305. The motion of an inferior planet is direct in passing through 
its superior conjunction, and retrograde in passing through its infe- 
rior conjunction. Thus Venus, while going from B through D to 
A, (Fig. 60,) moves in the order of the signs, or from west to east, 
and would appear to traverse the celestial vault B'S'A' from right 
to left ; but in passing from A through C to B, her course would 
be retrograde, returning on the same arc from left to right. If 


the earth were at rest, therefore, (and the sun, of course, at rest,) 
the inferior planets would appear to oscillate backwards and for- 
wards across the sun. But, it must be recollected, that the earth 
is moving in the same direction with the planet, as respects the 
signs, but with a slower motion. This modifies the motions of the 
planet, accelerating it in the superior and retarding it in the infe- 
rior conjunctions. Thus in figure 60, Venus while moving through 
BDA would seem to move in the heavens from B' to A' were the 
earth at rest ; but meanwhile the earth changes its position from 
E to E', by which means the planet is not seen at A' but at A", 
being accelerated by the arc A' A" in consequence of the earth's 
motion. On the other hand, when the planet is passing through 
its inferior conjunction ACB, it appears to move backwards in the 
heavens from A' to B' if the earth is at rest, but from A' to B" 
if the earth has in the mean time moved from E to E', being re- 
tarded by the arc B'B". Although the motions of the earth have 
the effect to accelerate the planet in the superior conjunction, and 
to retard it in the inferior, yet, on account of the greater distance, 
the apparent motion of the planet is much slower in the superior 
than in the inferior conjunction. 

306. When passing from the superior to the inferior conjunction, 
or from the inferior to the superior conjunction, through the greatest 
elongations, the inferior planets are stationary. 

If the earth were at rest, the stationary points would be at the 
greatest elongations as at A and B, for then the planet would be 
moving directly towards or from the earth, and would be seen for 
some time in the same place in the heavens ; but the earth itself 
is moving nearly at right angles to the line of the planet's motion, 
that is, the line which is drawn from the earth to the planet through 
the point of greatest elongation ; hence a direct motion is given 
to the planet by this cause. When the planet, however, has passed 
this line, by its superior velocity it soon overcomes this tendency 
of the earth to give it a relative motion eastward, and becomes 
retrograde as it approaches the inferior conjunction. Its stationary 
point obviously lies between its place of greatest elongation, and 
the place where its motion becomes retrograde. Mercury is sta- 


tionary at an elongation of from 15 to 20 from the sun; and 
Venus at about 29.* 

307. Mercury and Venus exhibit to the telescope phases similar to 
those of the moon. 

When on the side of their inferior conjunctions, these planets 
appear horned, like the moon in her first and last quarters ; and 
when on the side of their superior conjunctions, they appear gib- 
bous. At the moment of superior conjunction, the whole enlight- 
ened orb of the planet is turned towards the earth, and the appear- 
ance would be that of the full moon, but the planet is too near the 
sun to be commonly visible. 

These different phases show that these bodies are opake, and 
shine only as they reflect to us the light of the sun ; and the same 
remark applies to all the planets. 

308. The distance of an inferior planet from the sun, may be 
found by observations at the time of its greatest elongation. 

Thus if E be the place of the earth, and B that of Venus at the 
time of her greatest elongation, the angle SEE will be known, 
being a right angle. Also the angle SEE is known from observa- 
tion. Hence the ratio of SB to SE becomes known ; or, since SE 
is given, being the distance of the earth from the sun, SB the radius 
of the orbit of the planet is determined. If the orbits were both 
circles, this method would be very exact ; but being elliptical, we 
obtain the mean value of the radius SB by observing its greatest 
elongation in different parts of its orbit, f 

309. The orbit of Mercury is the most eccentric, and the most 
inclined of all the planets ; J while that of Venus varies but little 
from a circle, and lies much nearer to the ecliptic. 

The eccentricity of the orbit of Mercury is nearly } its semi- 
major axis, while that of Venus is only yjj ; the inclination of 
Mercury's orbit is 7, while that of Venus is less than 3|. Mer- 
cury, on account of his different distances from the earth, varies 

* Herschel, p. 242. -Woodhouse, 557. t Herschel, p. 239. 

t The new planets are of course excepted. Baily's Tables. 


much in his apparent diameter, which is only 5" in the apogee, 
but 12" in the perigee. 

310. The most favorable time for determining the sidereal revo- 
lution of a planet, is when its conjunction takes place at one of 
its nodes ; for then the sun, the earth, and the planet, being in the 
same straight line, it is referred to its true place in the heavens, 
whereas, in other positions, its apparent place is more or less 
affected by perspective. 

311. An inferior planet is brightest at a certain point between 
its greatest elongation and inferior conjunction. 

Its maximum brilliancy would happen at the inferior conjunc- 
tion, (being then nearest to us,) if it shined by its own light ; 
but in that position, its dark side is turned towards us. Still, its 
maximum cannot be when most of the illuminated side is towards 
us ; for then, being at the superior conjunction, it is at its greatest 
distance from us. The maximum must therefore be somewhere 
between the two. Venus gives her greatest light when about 40 
from the sun. 

312. Mercury and Venus both revolve on their axes, in nearly the 
same time with the earth. 

The diurnal period of Mercury is 24h. 5m. 28s., and that of 
Venus 23h. 21m. 7s. The revolutions on their axes have been 
determined by means of some spot or mark seen by the telescope, 
as the revolution of the sun on his axis is ascertained by means of 
his spots. 

313. Venus is regarded as the most beautiful of the planets, and 
is well known as the morning and evening star. The most ancient 
nations did not indeed recognize the evening and morning star as 
one and the same body, but supposed they were different planets, 
and accordingly gave them different names, calling the morning 
star Lucifer, and the evening star Hesperus. At her period of 
greatest splendor, Venus casts a shadow, and is sometimes visible 
in broad daylight. Her light is then estimated as equal to that of 


twenty stars of the first magnitude.* At her period of greatest 
elongation, Venus is visible from three to four hours after the set- 
ting or before rising of the sun. 

314. Every eight years, Venus forms lier conjunctions with the 
sun in the same part of the heavens. 

For, since the synodical period of Venus is 584 days, and her 
sidereal period 224.7, 

224.7 : 360:: 584 : 935.6=the arc of longitude described by 
Venus between thefirst and second conjunctions. Deducting 720, 
or two entire circumferences, the remainder, 215.6, shows how 
far the place of the second conjunction is in advance of the first. 
Hence, in five synodical revolutions, or 2920 days, the same point 
must have advanced 215.6x5=1078, which is nearly three 
entire circumferences, so that at the end of five synodical revolu- 
tions, occupying 2920 days, or 8 years, the conjunction of Venus 
takes place nearly in the same place in the heavens as at first. 

Whatever appearances of this planet, therefore, arise from its 
positions with respect to the earth and the sun, they are repeated 
every eight years in nearly the same form. 


315. The Transit of Mercury or Venus, is its passage across the 
sun's disk, as the moon passes over it in a solar eclipse. 

As a transit takes place only when the planet is in inferior con- 
junction, at which time her motion is retrograde (Art. 305,) it is 
always from left to right, and the planet is seen projected on the 
solar disk in a black round spot. Were the orbits of the inferior 
planets coincident with the plane of the earth's orbit, a transit 
would occur to some part of the earth at every inferior conjunc- 
tion. But the orbit of Venus makes an angle of 3 with the 
ecliptic, and Mercury an angle of 7 ; and, moreover, the apparent 
diameter of each of these bodies is very small, both of which cir- 
cumstances conspire to render a transit a comparatively rare 
occurrence, since it can happen only when the sun, at the time of 

* Francoeur, Uranography, p. 125. 


an inferior conjunction, chances to be at or extremely near the 
planet's node. The nodes of Mercury lie in longitude 46 and 
226, points which the sun passes through in May and November. 
It is only in these months, therefore, that transits of Mercury can 
occur. For a similar reason, those of Venus occur only in June 
and December. Since, however, the nodes of both planets have 
a small retrograde motion, the months in which transits occur will 
change in the course of ages. 

316. The intervals between successive transits, may be found in 
the following manner. The formula which gives the synodical 

period (Art. 304,) is 8=^7=, where S denotes the period, T the 

sidereal revolution of the earth, and T 7 that of the planet. If we 
now represent by m the number of synodical periods of the sun* 
in the required interval, and by n the number of synodical periods 
of the planet ; then, since the number of periods in each case is in- 
versely as the time of one, we have, T : - =^: : n : m .*. T __riv 
In the case of Mercury, whose sidereal period is 87.969 days, while 


that of the earth is, 365.256 days, = ~^^^ tnat * s > a ^ ter tne 

earth has revolved 87969 times, (or after this number of years,) 
Mercury will have revolved just 277287, and the two bodies will 
be together again at the place where they started. But as periods 
of such enormous length do not fall within the. observation of 
man, let us search for smaller numbers having nearly the same 
ratio. Now, 

87969 : 365256 : : 1 : 4 (nearly.) 

This shows that in one year Mercury will have made 4 revo- 
lutions and of another ; so that, when the sun returns to the same 
node, Mercury will be more than 60 in advance of it ; conse- 
quently, no transit can take place after an interval of one year. 
But, by making trial of 2, 3, 4, &c. years, we shall find a nearer 
approximation at the end of 6 years ; for, 

* That is, the time in which the sun returns again to the planet's node, which is ob- 
viously after one year. 



87969 : 365265 :: 6 : 25 T 1 T . In 6. years, therefore, Mercury 
will fall short of reaching the node by only T V of a revolution, or 
about 33. In 13 years the chance of meeting will be much 
greater, for in this period the earth will have made 1 3 and Mer- 
cury 54 revolutions. The numbers 33 and 137, 46 and 191, afford 
a still nearer approximation.* 

317. In a similar manner, transits of Venus are probable after 
8, 227, 235, and 243 years. Since Venus returns to her conjunc- 
tion at nearly the same point of her orbit, after 8 years, (Art. 314,) 
it frequently happens that a transit takes place after an interval 
of 8 years. But at that time Venus is so far from her node, that 
her latitude amounts to from 20' to 24'. Still she may possibly 
come within the sun's disk as she passes by him ; for suppose at 
the preceding transit her latitude was 10' on one side of the node 
and is now 10' on the other side, this being less than the sun's 
semi-diameter, a transit may occur 8 years after another. Thus 
transits of Venus took place in 1761 and 1769. But in 16 years 
the latitude changes from 40' to 48', and Venus could not reach 
any part of the solar disk in her inferior conjunction. 

From the above series we should infer that another transit 
could not take place under 227 years ; but since there are two 
nodes, the chance is doubled, so that a transit may occur at the 
other node in half that interval, or in about 113 years. If, at the 
occurrence of the first transit, Venus had passed her node, the 
next transit at the other node will happen 8 years before the 113 
are completed ; or if she had not reached the node, it will happen 
8 years later. Hence, after two transits have occurred within 8 
years, another cannot be expected before 105, 113, or 121 years. 
Thus, the next transit will happen in 1874=1769+105; also in 

318. The great interest attached by astronomers to a transit of 
Venus, arises from its furnishing the most accurate means in our 
power of determining the sun's horizontal parallax, an element 
of great importance, since it leads us to a knowledge of the distance 

* This series may readily be obtained by the method of Continued Fractions. See 
Davies's Bourdon's Algebra. 


of the earth from the sun, and consequently, by the application 
of Kepler's law, (Art. 183,) of the distances of all the other planets. 
Hence, in 1769, great efforts were made throughout the civilized 
world, under the patronage of different governments, to observe 
this phenomenon under circumstances the most favorable for de- 
termining the parallax of the sun. 

The method of finding the parallax of a heavenly body described 
in article 85, cannot be relied on to a greater degree of accuracy 
than 4". In the case of the moon, whose greatest parallax amounts 
to about 1, this deviation from absolute accuracy is not material ; 
but it amounts to nearly half the entire parallax of the sun. 

319. If the sun and Venus were equally distant from us, they 
would be equally affected by parallax as viewed by spectators in 
different parts of the earth, and hence their relative situation would 
not be altered by it ; but since Venus, at the inferior conjunction, 
is only about one third as far off as the sun, her parallax is propor- 
tionally greater, and therefore spectators at distant points will see 
Venus projected on different parts of the solar disk, and as the 
planet traverses the disk, she will appear to describe chords of dif- 
ferent lengths, by means of which the duration of the transit may 
be estimated at different places. The difference in the duration 
of the transit does not amount to many minutes ; but to make it 
as large as possible very distant places are selected for observation. 
Thus in the transit of 1769, among the places selected, two of the 
most favorable were Wardhuz in Lapland, and Oteheite,* one of 
the South Sea Islands. 

The principle on which the sun's horizontal parallax is estimated 
from the transit of Venus, may be illustrated as follows : Let E 
(Fig. 61,) be the earth, V Venus, and S the sun. Suppose A, B, 
two spectators at opposite extremities of that diameter of the earth 
which is perpendicular to the ecliptic. The spectator at A will 
see Venus on the sun's disk at a, and the spectator at B will see 
Venus at b ; and since AV and BV may be considered as equal 
to each other, as also V6 and Va, therefore the triangles AVB and 
Vab are similar to each other, and AV : Va : : AB : db. But the 
ratio of AV to Va is known, (Art. 308) ; hence, the ratio of AB to 

* Now written Tahiti. 


db is known, and when the angular value of ab as seen from the 
earth, is found, that of AB becomes known, as seen from the sun ;* 
and half AB, or the semi-diameter of the earth as seen from the 

Fig. 61. 

sun, is the sun's horizontal parallax. To find the apparent diameter 
of db, we have only to find the breadth of the space between the 
two chords. Now, we can ascertain the value of each chord by 
the time occupied in describing it, since the motions of Venus and 
those of the sun are accurately known from the tables. Each 
chord being double the sine of half the arc cut off by it, therefore 
the sine of half the arc and of course the versed sine becomes 
known, and the difference of the two versed sines is the breadth 
of the zone in question. There are many circumstances to be 
taken into the account in estimating, from observations of this 
kind, the sun's horizontal parallax ; but the foregoing explanation 
may be sufficient to give the learner an idea of the general princi- 
ples of this method. The appearance of Venus on the sun's disk, 
being that of a well defined black spot, and the exactness with 
which the moment of external or internal contact may be deter- 
mined, are circumstances favorable to the exactness of the result ; 
and astronomers repose so much confidence in the estimation of 
the sun's horizontal parallax as derived from the observations on 
the transit of 1769, that this important element is thought to be 
ascertained within T V of a second. The general result of all these 
observations give the sun's horizontal parallax 8."6, or more ex- 
actly, 8."5776.f 

* If, for example, ab is 2$ times AB, (which is nearly the fact,) then if AB were on 
the sun instead of on the earth, it would subtend an angle at the eye equal to ^ of ab. 

But if viewed from the sun, the distance being the same, its apparent diameter must be 
the same, 
t Delambre, t. 2. Vince's Complete Syst. vol. 1. Woodhouse, p. 754. Herschel, p. 243, 


320. During the transits of Venus over the sun's disk in 1761 
and 1769, a sort of penumbral light was observed around the 
planet by several astronomers, which was thought to indicate an 
atmosphere. This appearance was particularly observable while 
the planet was coming on and going off the solar disk. The total 
immersion and emersion were not instantaneous ; but as two drops 
of water, when about to separate, form a ligament between them, 
so there was a dark shade stretched out between Venus and the 
sun, and when the ligament broke, the planet seemed to have got 
about an eight part of her diameter from the limb of the sun.* 
The various accounts of the two transits abound with remarks like 
these, which indicate the existence of an atmosphere about Venus 
of nearly the density and extent of the earth's atmosphere. Similar 
proofs of the existence of an atmosphere around this planet, are 
derived from appearances of twilight. 



321. THE Superior planets are distinguished from the Inferior, 
by being seen at all distances from the sun from to 180. 
Having their orbits exterior to that of the earth, they of course 
never come between us and the sun, that is, they never have any 
inferior conjunction like Mercury and Venus, but they are some- 
times seen in superior conjunction, and sometimes in opposition. 
Nor do they, like the inferior planets, exhibit to the telescope dif- 
ferent phases, but, with a single exception, they always present 
the side that is turned towards the earth fully enlightened. This 
is owing to their great distance from the earth ; for were the spec- 
tator to stand upon the sun, he would of course always have the 
illuminated side of each of the planets turned towards him ; but, 
so distant are all the superior planets except Mars, that they are 

* Edinb. Encyc. Art. Astronomy. 


viewed by us very nearly in the same manner as they would be if 
we actually stood on the sun. 

322. Mars is a small planet, his diameter being only about half 
that of the earth, or 4200 miles. He also, at times, comes nearer to 
the earth than any other planet except Venus. His mean distance 
from the sun is 142,000,000 miles; but his orbit is so eccentric 
that his distance varies much in different parts of his revolution. 
Mars is always very near the ecliptic, never varying from it 2. 
He is distinguished from all the other planets by his deep red color, 
and fiery aspect ; but his brightness and apparent magnitude vary 
much at different times, being sometimes nearer to us than at 
others, by the whole diameter of the earth's orbit, that is, by about 
190,000,000 of miles. When Mars is on the same side of the sun 
with the earth, or at his opposition, he comes within 47,000,000 
miles of the earth, and rising about the time the sun sets surprises 
us by his magnitude and splendor; but when he passes to the 
other side of the sun to his superior conjunction, he dwindles to 
the appearance of a small star, being then 237,000,000 miles from 
us. Thus, let M (Fig. 62,) represent Mars in opposition, and M' 

Fig. 62. 

in the superior conjunction. It is obvious that in the former situa- 
tion, the planet must be nearer to the earth than in the latter by 
the whole diameter of the earth's orbit. 


323. Mars is the only one of the superior planets which exhibits 
phases. When he is towards the quadratures at Q or Q', it is 
evident from the figure that only a part of the circle of illumina- 
tion is turned towards the earth, such a* portion of the remoter 
part of it being concealed from our view as to render the form 
more or less gibbous. 

324. "When viewed with a powerful telescope, the surface of 
Mars appears diversified with numerous varieties of light and 
shade. The region around the poles is marked by white spots, 
which vary their appearance with the changes of seasons in the 
planet. Hence Dr. Herschel conjectured that they were owing 
to ice and snow, which alternately accumulates and melts, accord- 
ing to the position of each pole with respect to the sun.* It has 
been common to ascribe the ruddy light of this planet to an exten- 
sive and dense atmosphere, which was said to be distinctly indi- 
cated, by the gradual diminution of light observed in a star as it 
approached very near to the planet in undergoing an occultation ; 
but more recent observations afford no such evidence of an atmo- 

By observations on the spots we learn that Mars revolves on his 
axis in very nearly the same time with the earth, (24h. 39m. 21 8 .3); 
and that the inclination of his axis to that of the ecliptic is also 
nearly the same, being 30 18' 10".8.J 

325. As the diurnal rotation of Mars is nearly the same as that 
of the earth, we might expect a similar flattening at the poles, 
giving to the planet a spheroidal figure. Indeed the compression 
or ellipticity of Mars greatly exceeds that of the earth, being no 
less than T V of the equatorial diameter, while that of the earth is 
only ai T , (Art. 138.) This remarkable flattening of the poles of 
Mars has been supposed to arise from a great variation of density 
in the planet in different parts. 

326. JUPITER is distinguished from all the other planets by his 
vast magnitude. His diameter is 89,000 miles, and his volume 

Phil. Trans. 1784. t Sir James South, Phil. Trans. 1833. 

t Baily's Tables, p. 29. Ed. Encyc. Art. Astronomy. 


1280 times that of the earth. His figure is strikingly spheroidal, 
the equatorial being larger than the polar diameter in the propor- 
tion of 107 to 100. (See Frontispiece, Fig. 4.) Such a figure 
might naturally be expected from the rapidity of his diurnal rota- 
tion, which is accomplished in about 10 hours. A place on the 
equator of Jupiter must turn 27 times as fast as on the terrestrial 
equator. The distance of Jupiter from the sun is nearly 490,000,000 
miles, and his revolution around the sun occupies nearly 12 

327. The view of Jupiter through a good telescope, is one of 
the most magnificent and interesting spectacles in astronomy. 
The disk expands into a large and bright orb like the full moon ; 
the spheroidal figure which theory assigns to revolving spheres, is 
here palpably exhibited to the eye ; across the disk, arranged in 
parallel stripes, are discerned several dusky bands, called belts ; 
and four bright satellites, always in attendance, but ever varying 
their positions, compose a splendid retinue. Indeed, astronomers 
gaze with peculiar interest on Jupiter and his moons as affording 
a miniature representation of the whole solar system, repeating on 
a smaller scale, the same revolutions, and exemplifying, in a man- 
ner more within the compass of our observation, the same laws as 
regulate the entire assemblage of sun and planets. (See Fig. 63.) 

328. The Belts of Jupiter, are variable in their number and di- 
mensions. With the smaller telescopes, only one or two are sean 
across the equatorial regions ; but with more powerful instruments, 
the number is increased, covering a great part of the whole disk. 
Different opinions have been entertained by astronomers respect- 
ing the cause of the belts ; but they have generally been regarded 
as clouds formed in the atmosphere of the planet, agitated by 
winds, as is indicated by their frequent changes, and made to as- 
sume the form of belts parallel to the equator by currents that cir- 
culate around the planet like the trade winds and other currents 
that circulate around our globe.* Sir John Herschel supposes 
that the belts are not ranges of clouds, but portions of the planet 
itself brought into view by the removal of clouds and mists 

* Ed. Encyc. Art. Astronomy. 


that exist in the atmosphere of the planet through which are open- 
ings made by currents circulating around Jupiter.* 

329. The Satellites of Jupiter may be 'seen with a telescope of 
very moderate powers. Even a common spy-glass will enable us 
to discern them. Indeed one or two of them have been occasion- 
ally seen with the naked eye. In the largest telescopes, they sev- 
erally appear as bright as Sirius. With such an instrument the 
view of Jupiter with his moons and belts is truly a magnificent 
spectacle, a world within itself. As the orbits of the satellites do 
not deviate far from the plane of the ecliptic, and but little from 
the equator of the planet, they are usually seen in nearly a straight 
line with each other extending across the central part of the disk. 
(See Frontispiece.) 

330. Jupiter's satellites are distinguished from one another by 
the denominations of first, second* third, and fourth, according to 
their relative distances from Jupiter, the first being that which is 
nearest to him. Their apparent motion is oscillatory, like that of 
a pendulum, going alternately from their greatest elongation on 
one side to their greatest elongation on the other, sometimes in a 
straight line, and sometimes in an elliptical curve, according to the 
different points of view in which we observe them from the earth. 
They are sometimes stationary ; their motion is alternately direct 
and retrograde ; and, in short, they exhibit in miniature all the 
phenomena of the planetary system. Various particulars of the 
system are exhibited in the following table. The distances are 
given in radii of the primary. 



Mean Distance. 

Sidereal Revolution. 

' 4 



Id. 18h. 28m. 
3 13 14 
7 3 43 
16 16 32 

Hence it appears, first, that Jupiter's satellites are all somewhat 
larger than the moon, (except the second, which is very nearly 
the size of the moon,) and the third the largest of the whole, but the 

* Herschel's Astron. p. 266 


diameter of the latter is only about -fa part of that of the primary ; 
secondly, that the distance of the innermost satellite from the 
planet is three times his diameter, while that of the outermost 
satellite is nearly fourteen times his diameter ; thirdly, that the 
first satellite completes its revolution around the primary in one 
day and three fourths, while the fourth satellite requires nearly 
sixteen and three fourths days. 

331. The orbits of the satellites are nearly or quite circular, and 
deviate but little from the plane of the planet's equator, and of 
course are but slightly inclined to the plane of his orbit. They 
are, therefore, in a similar situation with respect to Jupiter as the 
moon would be with respect to the earth if her orbit nearly coin- 
cided with the ecliptic, in which case she would undergo an eclipse 
at every opposition. 

332. The eclipses of Jupiter's satellites, in their general concep- 
tion, are perfectly analogous to those of the moon, but in their de- 
tail they differ in several particulars. Owing to the much greater 
distance of Jupiter from the sun, and its greater magnitude, the 
cone of its shadow is much longer and larger than that of the 
earth, (Art. 246.) On this account, as well as on account of the 
little inclination of their orbits to that of their primary, the three 
inner satellites of Jupiter pass through the shadow, and are totally 
eclipsed at every revolution. The fourth satellite, owing to the 
greater inclination of its orbit, sometimes though rarely escapes 
eclipse, and sometimes merely grazes the limits of the shadow or 
suffers a partial eclipse.* These eclipses, moreover, are not seen 
as is the case with those of the moon, from the center of their mo- 
tion, but from a remote station, and one whose situation with re- 
spect to the line of the shadow is variable. This, of course, makes 
no difference in the times of the eclipses, but a very great one in 
their visibility, and in their apparent situations with respect to the 
planet at the moment of their entering or quitting the shadow. 

333. The eclipses of Jupiter's satellites present some curious 
phenomena, which will be understood from the following diagram. 

* Sir J. Herschel, Ast. p. 276. 


Let A, B, C, D, (Fig. 63,) represent the earth in different parts of 
its orbit ; J, Jupiter in his orbit MN, surrounded by his four satel- 
lites, the orbits of which are marked 1, 2, 3, 4. At a the first 
satellite enters the shadow of the planet, and emerges from it at 
6, and advances to its greatest elongation at c. Since the shadow 
is always opposite to the sun, only the immersion of a satellite 
will be visible to the earth while the earth is somewhere between 

Fig. 63. 

C and A, that is, while the earth is passing from the position 
where it has the planet in superior conjunction, to that where it 
has the planet in opposition ; for while the earth is in this situation, 
the planet conceals from its view the emersion, as is evident from 
the direction of the visual rayfd. For a similar reason the emer- 
sion only is visible while the earth passes from A to C, or from 
the opposition to the superior conjunction. In other words, when 
the earth is to the westward of Jupiter, only the immersions of a 
satellite are visible ; when the earth is to the eastward of Jupiter, 
only the emersions are visible. This, however, is strictly true only 
of the first satellite ; for the third and fourth, and sometimes even 
the second, owing to their greater distances from Jupiter, occa- 
sionally disappear and reappear on the same side of the disk. 
The reason why they should reappear on the same side of the 
disk, will be understood from the figure. Conceive the whole sys- 
tem of Jupiter and his satellites as projected on the more distant 
concave sphere, by lines drawn, like/cZ, from the observer on the 
earth at D through the planet and each of the satellites ; then it is 
evident that the remoter parts of the shadow where the exterior sat- 
ellites traverse it, will fall to the westward of the planet, and of 


course these satellites as they emerge from the shadow will be pro- 
jected to a point on the same side of the disk as the point of their 
immersion. The same mode of reasoning will show that when 
the earth is to the eastward of the planet, the immersions and 
emersions of the outermost satellites will be both seen on the east 
side of the disk. When the earth is in either of the positions C 
or A, that is, at the superior conjunction or opposition of the planet, 
both the immersions and emersions take place behind the planet, 
and the eclipses occur close to the disk. 

334. When one of the satellites is passing between Jupiter and 
the sun, it casts a shadow upon its primary, which is seen by the 
telescope travelling across the disk of Jupiter, as the shadow of the 
moon would be seen to traverse the earth by a spectator favor- 
ably situated in space. When the earth is to the westward of Ju- 
piter, as at D, the shadow reaches the disk of the planet, or is seen 
on the disk, before the satellite itself reaches it. For the satellite 
will not enter on the disk, until it comes up to the line fd at d, a 
point which it reaches later than its shadow reaches the same line. 
After the earth has passed the opposition, as at B, then the satel- 
lite will reach the visual ray at d sooner than the shadow, and 
of course be sooner projected on the disk. In the transits of Ju- 
piter's satellites, which with very powerful telescopes may be ob- 
served with great precision, the satellite itself is sometimes seen on 
the disk as a bright spot, if it chances to be projected upon one of 
the belts. Occasionally, also, it is seen as a dark spot, of smaller 
dimensions than the shadow. This curious fact has led to the 
conclusion, that certain of the satellites have sometimes on their 
own bodies or in their atmospheres, obscure spots of great extent.* 

335. A very singular relation subsists between the mean motions 
of the three first satellites of Jupiter. The mean longitude of thp 
first satellite, minus three times that of the second, plus twice that of 
the third, always equals 180 degrees. A curious consequence of 
this relation is, that the three satellites can never be all eclipsed at 
the same time ; for then their longitudes would be equal, and of 

* Sir J. HerscheL 


course the sum of their longitudes would be nothing.* It will be 
remarked, that these phenomena are such as would present them- 
selves to a spectator on Jupiter, and not to a spectator on the 

336. The eclipses of Jupiter's satellites have been studied with 
great attention by astronomers, on account of their affording one 
of the easiest methods of determining the longitude. On this 
subject Sir J. Herschel remarks :f The discovery of Jupiter's 
satellites by Galileo, which was one of the first fruits of the inven- 
tion of the telescope, forms one of the most memorable epochs in 
the history of astronomy. The first astronomical solution of the 
great problem of " the longitude," the most important problem 
for the interests of mankind that has ever been brought under the 
dominion of strict scientific principles, dates immediately from 
their discovery. The final and conclusive establishment of the 
Copernican system of astronomy, may also be considered as refer- 
able to the discovery and study of this exquisite miniature system, 
in which the laws of the planetary motions, as ascertained by 
Kepler, and especially that which connects their periods and dis- 
tances, were speedily traced, and found to be satisfactorily main- 

337. The entrance of one of Jupiter's satellites into the shadow 
of the primary being seen like the entrance of the moon into the 
earth's shadow, at the same moment of absolute time, at all 
places where the planet is visible, and being wholly independent of 
parallax ; being, moreover, predicted beforehand with great accu- 
racy for the instant of its occurrence at Greenwich, and given in 
the Nautical Almanac ; this would seem to be one of those events 
(Art. 273,) which are peculiarly adapted for finding the longitude. 
It must be remarked, however, that the extinction of light in the 
satellite at its immersion, and the recovery of its light at its emer- 
sion, are not instantaneous, but gradual ; for the satellite, like the 
moon, occupies some time in entering into the shadow or in 
emerging from it, which occasions a progressive diminution or in- 

* Biot, Ast. Phy. t Elements of Ast. p. 279. 


crease of light. The better the light afforded by the telescope 
with which the observation is made, the later the satellite will be 
seen at its immersion, and the sooner at its emersion.* In noting 
the eclipses even of the first satellite, the time must be considered 
as uncertain to the amount of 20 or 30 seconds ; and those of the 
other satellites involve still greater uncertainty. Two observers, 
in the same room, observing with different telescopes the same 
eclipse, will frequently disagree in noting its time to the amount 
of 15 or 20 seconds ; and the difference will be always the same 

Better methods, therefore, of finding the longitude are now 
employed, although the facility with which the necessary observa- 
tions can be made, and the little calculation required, still render 
this method eligible in many cases where extreme accuracy is not 
important. As a telescope is essential for observing an eclipse of 
one of the satellites, it is obvious that this method cannot be prac- 
ticed at sea. 

338. The grand discovery of the progressive motion of light, 
was first made by observations on the eclipses of Jupiter's satel- 
lites. In the year 1675, it was remarked by Roemer, a Danish 
astronomer, on comparing.together observations of these eclipses 
during many successive years, that they take place sooner by 
about sixteen minutes (16m. 26 s . 6) when the earth is on the 
same side of the sun with the planet, than when she is on the op- 
posite side. This difference he ascribed to the progressive motion 
of light, which takes that time to pass through the diameter of the 
earth's orbit, making the velocity of light about 192,000 miles per 
second. So great a velocity startled astronomers at first, and pro- 
duced some degree of distrust of this explanation of the phenome- 
non ; but the subsequent discovery of the aberration of light (Art. 
195,) led to an independent estimation of the velocity of light 
with almost precisely the same result. 

339. SATURN comes next in the series as we recede from the 

* This is the reason why observers are directed in the Nautical Almanac to use tele, 
copes of a certain power. 
t Woodhouse, p. 840. 

SATURN. 199 

sun, and has, like Jupiter, a system within itself, on a scale of great 
magnificence. In size it is, next to Jupiter, the largest of the 
planets, being 79,000 miles in diameter, or about 1,000 times as 
large as the earth. It has likewise belts on its surface and is at- 
tended by seven satellites. But a still more wonderful appendage 
is its Ring, a broad wheel encompassing the planet at a great dis- 
tance from it. We have already intimated that Saturn's system 
is on a grand scale. As, however, Saturn is distant from us nearly 
900,000,000 miles, we are unable to obtain the same clear and 
striking views of his phenomena as we do of the phenomena of 
Jupiter, although they really present a more wonderful mechanism. 

340. Saturn's ring, when viewed with telescopes of a high 
power, is found to consist of two concentric rings,* separated from 
each other by a dark space. (See Frontispiece.) Although this 
division of the rings appears to us, on account of our immense dis- 
tance, as only a fine line, yet it is in reality an interval of not less 
than about 1800 miles. The dimensions of the whole system are 
in round numbers, as follows :f 


Diameter of the planet, .... 79,000 
From the surface of the planet to the inner ring, 20,000 
Breadth of the inner ring, : . . . . 17,000 
Interval between the rings, .. . . 1,800 

Breadth of the outer ring, .... 10,500 
Extreme dimensions from outside to outside, 176,000 
The figure represents Saturn as it appears to a powerful tele- 
scope, surrounded by its rings, and having its body striped with 
dark belts, somewhat similar but broader and less strongly marked 
than those of Jupiter, and owing doubtless to a similar cause. 
That the ring is a solid opake substance, is shown by its throwing 
its shadow on the body of the planet on the side nearest the sun 
and on the other side receiving that of the body. From the par- 
allelism of the belts with the plane of the ring, it may be conjec- 
tured that the axis of rotation of the planet is perpendicular to 

* It is said that several additional divisions of the ring have been detected. (Kater, 
Ast. Trans, iv. 383.) t Prof. Struve, Mem. Ast. Soc., 3. 301. 


th it plane ; and this conjecture is confirmed by the occasional 
a] pearance of extensive dusky spots on its surface, which when 
watched indicate a rotation parallel to the ring in lOh. 29m. 17s. 
This motion, it will be remarked, is nearly the same with the diur- 
nal motion of Jupiter, subjecting places on the equator of the 
planet to a very swift revolution, and occasioning a high degree 
of compression at the poles, the equatorial being to the polar di- 
ameter in the high ratio of 11 to 10. But it is remarkable that the 
globe of Saturn appears to be flattened at the equator as well as 
at the poles. The polar compression extends to a great distance 
over the surface of the planet, and the greatest diameter is that of 
the parallel of 43 of latitude. The dis'k of Saturn, therefore, re- 
sembles a square of which the four corners have been rounded off.* 
It requires a telescope of high magnifying powers and a strong 
light to give a full and striking view of Saturn with his rings and 
belts and satellites ; for we must bear in mind that at the distance 
of Saturn one second of angular measurement corresponds to 4,000 
miles, a space equal to the semi-diameter of our globe. But with 
a telescope of moderate powers, the leading phenomena of the 
ring itself may be observed. 

341. Saturn 1 s ring, in its revolution around the sun, always re- 
mains parallel to itself. 

If we hoM opposite to> the eye a circular ring or disk like a 
piece of coin, it will appear as a complete circle when it is at right 
angles to the axis of vision, but when oblique to that axis it will 
be projected into an ellipse more and more flattened as its obliquity 
is increased, until, when its plane coincides with the axis of vision, 
it is projected into a straight line. Let us place on the table a 
lamp to represent the sun, and holding the ring at a certain dis- 
tance inclined a little towards the lamp, let us carry it round the 
lamp, always keeping it parallel to itself. During its revolution it 
will twice present its edge to the lamp at opposite points, and 
twice at places 90 distant from those points, it will present its 
broadest face towards the lamp. At intermediate points, it will 
exhibit an ellipse more or less open, according as it is nearer one 

* Sir W. Herschel, Phil. Tr. 1806, Part II. 

SATURN. 201 

or the other of the preceding positions. It will be seen also that 
in one half of the revolution the lamp shines on one side of the 
ring, and in the other half of the revolution on the other side. 
Such would be the successive appearances of Saturn's ring to a 
spectator on the sun ; and since the earth is, in respect to so dis- 
tant a body as Saturn, very near the sun, those appearances are 
presented to us in nearly the same manner as though we viewed 
them from the sun. Accordingly, we sometimes see Saturn's ring 
under the form of a broad ellipse, which grows continually more 
and more acute until it passes into a line, and we either lose sight 
of it altogether, or with the aid of the most powerful telescopes, 
we see it as a fine thread of light drawn across the disk and pro- 
jecting out from it on each side. As the whole revolution occupies 
30 years, and the edge is presented to the sun twice in the revolu- 
tion, this last phenomenon, namely, the disappearance of the ring, 
takes place every 15 years. 

342. The learner may perhaps gain a clearer idea of the fore- 
going appearances from the following diagram : 

Let A, B, C, &c. represent successive positions of Saturn and 
his ring in different parts of his orbit, while ab represents the 
orbit of the earth.* Were the ring when at C and G perpendicu- 

Fig. 64. 

lar to the line CG, it would be seen by a spectator situated at a 
or b a perfect circle, but being inclined to the line of vision 28 4', 

*.It may be remarked by the learner, that these orbits are made so elliptical, not to 
represent the eccentricity of either the earth's or Saturn's orbit, but merely as the pro- 
jection of circles seen very obliquely. 



it is projected into an ellipse. This ellipse contracts in breadth 
as the ring passes towards its nodes at A and E, where it dwindles 
into a straight line. From E to G the ring opens again, becomes 
broadest at G, and again contracts till it becomes a straight line at 
A, and from this point expands till it recovers its original breadth 
at C. These successive appearances are all exhibited to a telescope 
of moderate powers. The ring is extremely thin, since the small- 
est satellite, when projected on it, more than covers it. The thick- 
ness is estimated at 100 miles. 

343. Saturn 9 s ring shines wholly by reflected light derived from 
the sun. This is evident from the fact, that that side only which 
is turned towards the sun is enlightened ; and it is remarkable, 
that the illumination of the ring is greater than that of the planet 
itself, but the outer ring is less bright than the inner. Although, as 
we have already remarked, we view Saturn's ring nearly as though 
we saw it from the sun, yet the plane of the ring produced may 
pass between the earth and the sun, in which case also the ring 
becomes invisible, the illuminated side being wholly turned from 
us. Thus, when the ring is approaching its node atE, a spectator 
at a would have, the dark side of the ring presented to him. The 
ring was invisible in 1833, and will be invisible again in 1847. 
At present (1841) it is the northern side of the ring that is seeij, 
but in 1855 the southern side will come into view. 

It appears, therefore, that there are three causes for the disap- 
pearance of Saturn's ring ; first, when the edge of the ring is pre- 
sented to the sun ; secondly, when the edge is presented to the 
earth ; and thirdly, when the unilluminated side is towards the 

344. Saturn's ring revolves in its own plane in about 10 hours, 
(lOh. 32m. 15 s . 4). La Place inferred this from the doctrine of 
universal gravitation. He proved that such a rotation was neces- 
sary, otherwise the matter of which the ring is composed would 
be precipitated upon its primary. He showed that in order to 
sustain itself, its period of rotation must be equal to the time of 
revolution of a satellite, circulating around Saturn at a distance 
from it equal to that of the middle of the ring, which period would 

SATURN. 203 

be about 10? hours. By means of spots in the ring Dr. Hersche. 
followed the ring in its rotation, and actually found its period to 
be the same as assigned by La Place, a coincidence which beau- 
tifully exemplifies the harmony of truth.*' 

345. Although the rings are very nearly concentric, yet recent 
measurements of extreme delicacy have demonstrated, that the 
coincidence is not mathematically exact, but that the center of 
gravity of the rings describes around that of the body a very 
minute orbit. This fact, unimportant as it may seem, is of the 
utmost consequence to the stability of the system of rings. Sup- 
posing them mathematically perfect in their circular form, and 
exactly concentric with the planet, it is demonstrable that they 
would form (in spite of their centrifugal force) a system in a state 
of unstable equilibrium, which the slightest external power would 
subvert not by causing a rupture in the substance of the rings 
but by precipitating them unbroken on the surface of the planet. ( 
The ring may be supposed of an unequal breadth in its different 
parts, and as consisting of irregular solids, whose common center 
of gravity does not coincide with the center of the figure. Were 
it not for this distribution of matter, its equilibrium would be de- 
stroyed by the slightest force, such as the attraction of a satellite, 
and the ring would finally precipitate itself upon the planet. J 

As the smallest difference of velocity between the planet and 
its rings must infallibly precipitate the rings upon the planet, 
never more to separate, it follows either that their motions in their 
common orbit round the sun, must have been adjusted to each 
other by an external power, with the minutest precision, or that 
the rings must have been formed about the planet while subject 
to their common orbitual motion, and under the full and free in- 
fluence of all the acting forces. 

The rings of Saturn must present a magnificent spectacle from 
those regions of the planet which lie on their enlightened sides, 
appearing as vast arches spanning the sky from horizon to hori- 
zon, and holding an invariable situation among the stars. On 
the other hand, in the region beneath the dark side, a solar eclipse 

Systeme du Monde, 1. iv. c. 8. t Sir J. Herschel. t La Place. 


of 15 years in duration, under their shadow, must afford (to our 
ideas) an inhospitable abode to animated beings, but ill compen- 
sated by the full light of its satellites. But we shall do wrong 
to judge of the fitness or unfitness of their condition from what 
we see around us, when, perhaps, the very combinations which 
convey to our minds only images of horror, may be in reality 
theatres of the most striking and glorious displays of beneficent 

346. Saturn is attended by seven satellites. Although bodies 
of considerable size, their great distance prevents their being vis- 
ible to any telescopes but such as afford a strong light and high 
magnifying powers. The outermost satellite is distant from the 
planet more than 30 times the planet's diameter, and is by far 
the largest of the whole. It is the only one of the series whose 
theory has been investigated further than suffices to verify Kep- 
ler's law of the periodic times, which is found to hold good here 
as well as in the system of Jupiter. It exhibits, like the satellites 
of Jupiter, periodic variations of light, which prove its revolution 
on its axis in the time of a sidereal revolution about Saturn. 
The next satellite in order, proceeding inwards, is tolerably con- 
spicuous ; the three next are very minute, and require pretty pow- 
erful telescopes to see them ; while the two interior satellites, 
which just skirt the edge of the ring, and move exactly in its 
plane, have never been discovered but with the most powerful 
telescopes which human art has yet constructed, and then only 
under peculiar circumstances. At the time of the disappearance 
of the rings (to ordinary telescopes) they were seen by Sir Wil- 
liam Herschel with his great telescope, projected along the edge 
of the ring, and threading like beads the thin fibre of light to which 
the ring is then reduced. Owing to the obliquity of the ring, and 
of the orbits of the satellites to that of their primary, there are no 
eclipses of the satellites, the two interior ones excepted, until near 
the time when the ring is seen edgewise.f 

347. URANUS is the remotest planet belonging to our system, 

* Sir J. Herschel. t Sir J. Herschel. 

SATBRN. 205 


and is rarely visible except to the telescope. Although his diam- 
eter is more than four times that of the earth, (35,112 miles,) yet 
his distance from the sun is likewise nineteen times as great as 
the earth's distance, or about 1,800,000,000 miles. His revolution 
around the sun occupies nearly 84 years, so that his position in 
the heavens for several years in succession is nearly stationary. 
His path lies very nearly in the ecliptic, being inclined to it less 
than one degree, (46' 28".44.) 

The sun himself when seen from Uranus dwindles almost to a 
star, subtending as it does an angle of only 1' 40" ; so that the 
surface of the sun would appear there 400 times less than it does 
to us. 

This planet was discovered by Sir William Herschel on the 
13th of March, 1781. His attention was attracted to it by the 
largeness of its disk in the telescope ; and finding that it shifted 
its place among the stars, he at first took it for a comet, but soon 
perceived that its orbit was not eccentric like the orbits of comets, 
but nearly circular like those of the planets. It was then recog- 
nized as a new member of the planetary system, a conclusion 
which has been justified by all succeeding observations. 

348. Uranus is attended by six satellites. So minute objects 
are they that they can be seen only by powerful telescopes. In- 
deed the existence of more than two is still considered as some- 
what doubtful.* These two, however, offer remarkable, and in- 
deed quite unexpected and unexampled peculiarities. Contrary 
to the unbroken analogy of the whole planetary system, the planes 
of their orbits are nearly perpendicular to the ecliptic, being inclined 
no less than 78 58' to that plane, and in these orbits their motions 
are retrograde ; that is, instead of advancing from west to east 
around their primary, as is the case with all the other planets and 
satellites, they move in the opposite direction.f With this excep- 
tion, all the 'motions of the planets, whether around their own axes, 
or around the sun, are from west to east. 

* A third satellite of Uranus, is said to have been recently seen at Munich. (Jouu 
Franklin Inst. xxiii, 29.) 
t Sir J. Herschel. 



349. THE commencement of the present century was rendered 
memorable in the annals of astronomy, by the discovery of four 
new planets between Mars and Jupiter. Kepler, from some 
analogy which he found to subsist among the distances of the 
planets from the sun, had long before suspected the existence of 
one at this distance ; and his conjecture was rendered more prob- 
able by the discovery of Uranus, which follows the analogy oi 
the other planets. So strongly, indeed, were astronomers im- 
pressed with the idea that a planet would be found between Mars 
and Jupiter, that in the hope of discovering it, an association was 
formed on the continent of Europe of twenty-four observers, who 
divided the sky into as many zones, one of which was allotted to 
each member of the association. The discovery of the first of 
these bodies was however made accidentally by Piazzi, an astron- 
omer of Palermo, on the first of January, 1801. It was shortly 
afterwards lost sight of on account of its proximity to the sun, 
and' was not seen again until the close of the year, when it was 
re-discovered in Germany. Piazzi called it Ceres in honor of the 
tutelary goddess of Sicily, and her emblem, the sickle ? , has been 
adopted as its appropriate symbol. 

The difficulty of finding Ceres induced Dr. Olbers, of Bremen, 
to examine with particular care all the small stars that lie near 
her path, as seen from the earth ; and while prosecuting these 
observations, in March, 1802, he discovered another similar body, 
very nearly at the same distance from the sun, and resembling the 
former in many other particulars. The discoverer gave to this 
second planet the name of Pallas, choosing for its symbol the 
lance $ , the characteristic of Minerva. 

350. The most surprising circumstance connected with the 
discovery of Pallas, was the existence of two planets at nearly the 
same distance from the sun, and apparently having a common node. 
On account of this singularity, Dr. Olbers was led to conjecture 
that Ceres and Pallas are only fragments of a larger planet, which 
had formerly circulated at the same distance, and been shattered 
by some internal convulsion. La Grange, a mathematician of the 


first eminence, investigated the forces that would be necessary to 
detach a fragment from a planet with a velocity that would cause 
it to describe such orbits as these bodies are found to have. The 
hypothesis suggested the probability that there might be other 
fragments, whose orbits, however they might differ in eccentricity 
and inclination, might be expected to cross the ecliptic at a com- 
mon point, or to have the same node. Dr. Olbers, therefore, pro- 
posed to examine carefully every month the two opposite parts 
of the heavens in which the orbits of Ceres and Pallas intersect 
one another, with a view to the discovery of other planets, which 
might be sought for in those parts with greater chance of success 
than in a wider zone, embracing the entire limits of these orbits. 
Accordingly, in 1804, near one of the nodes of Ceres and Pallas, 
a third planet was discovered. This was called Juno, and the 
character $ was adopted for its symbol, representing the starry 
sceptre of the queen of Olympus. Pursuing the same researches, 
in 1807, a fourth planet was discovered, to which was given the 
name of Vesta, and for its symbol the character fi was chosen, 
an altar surmounted with a censer holding the sacred fire. 

After this historical sketch, it will be sufficient to classify under 
a few heads the most interesting particulars relating to the New 

351. The average distance of these bodies from the sun is 
261,000,000 miles ; and it is remarkable that their orbits are very 
near together. Taking the distance of the earth from the sun for 
unity, their respective distances are 2.77, 2.77, 2.67, 2.37. 

As they are found to be governed, like the other members of 
the solar system, by Kepler's law, that regulates the distances and 
times of revolution, their periodical times are of course pretty 
nearly equal, averaging about 4 years. 

In respect to the inclination of their orbits, there is considerable 
diversity. The orbit of Vesta is inclined to the ecliptic only 
about 7, while that of Pallas is more than 34. They all there- 
fore have a higher inclination than the orbits of the old planets, 
and of course make excursions from the ecliptic beyond the limits 
of the Zodiac. 

The eccentricity of their orbits is also, in general, greater than 


that of the old planets ; and the eccentricities of the orbits of Pal- 
las and Juno exceed that of the orbit of Mercury. 

Their small size constitutes one of their most remarkable pecu- 
liarities. The difficulty of estimating the apparent diameter of 
bodies at once so very small and so far off, would lead us to ex- 
pect different results in the actual estimates. Accordingly, while 
Dr. Herschel estimates the diameter of Pallas at only 80 miles, 
Schroeter places it as high as 2,000 miles, or about the size 
of the moon. The volume of Vesta is estimated at only one fif- 
teen thousandth part of the earth's, and her surface is only about 
equal to that of the. kingdom of Spain.* These little bodies are 
surrounded by atmospheres of great extent, some of which are un- 
commonly luminous, and others appear to consist of nebulous mat- 
ter. These planets in general shine with a more vivid light than 
might be expected from their great distance and diminutive size 



352. WE have waited until the learner may be supposed to be 
familiar with the contemplation of the heavenly bodies, individu- 
ally, before inviting his attention to a systematic view of the 
planets, and of their motions around the sun. The time has now 
arrived for entering more advantageously upon this subject, than 
could have been done at an earlier period. 

There are two methods of arriving at a knowledge of the mo- 
tions of the heavenly bodies. One is to begin with the apparent, 
and from these to deduce the real motions ; the other is, to begin 
with considering things as they really are in nature, and then to 
inquire why they appear as they do. The latter of these methods 
is by far the more eligible ; it is much easier than the other, and 

* New Encyc. Brit., Art. Astronomy. 


proceeding from the less difficult to that which is more so, from 
motions that are very simple to such as are complicated, it- finally 
puts the learner in possession of the whole machinery of the heav- 
ens. We shall, in the first place, therefore, endeavor to introduce 
the student to an acquaintance with the simplest motions of the 
planetary system, and afterwards to conduct him gradually 
through such as are more complicated and difficult. 

353. Let us first of all endeavor to acquire an adequate idea of 
absolute space, such as existed before the creation of the world. 
We shall find it no easy matter to form a correct notion of infinite 
space ; but let us fix our attention, for some time, upon extension 
alone, devoid of every thing material, without light or life, and 
without bounds. Of such a space we could not predicate the 
ideas of up or down, east, west, north, or south, but all reference 
to our own horizon (which habit is the most difficult of all to 
eradicate from the mind) must be completely set aside. Into such 
a void we would introduce the SUN. We would contemplate this 
body alone, in the midst of boundless space, and continue to fix 
he attention upon this object, until we had fully settled its rela- 
tions to the surrounding void. The ideas of up and down would 
now present themselves, but as yet there would be nothing to sug- 
gest any notion of the cardinal points. We suppose ourselves 
next to be placed on the surface of the sun, and the firmament of 
stars to be lighted up. The slow revolution of the sun on his axis, 
would be indicated by a corresponding movement of the stars in 
the opposite direction ; and in a period equal to more than 27 of 
our days, the spectator would see the heavens perform a complete 
revolution around the sun, as he now sees them revolve around 
the earth once in 24 hours. The point of the firmament where 
no motion appeared, would indicate the position of one of the 
poles, which being called North, the other cardinal points would 
be immediately suggested. 

Thus prepared, we may now enter upon the consideration of 
the planetary motions. 

354. Standing on the sun, we see all the planets moving slowly 
around the celestial sphere, nearly in the same great high way, and 



in the same direction from west to east. They move, however, 
with very unequal velocities. Mercury makes very perceptible 
progress from night to night, like the moon revolving about the 
earth, his daily progress eastward being about one third as great as 
that of the moon, since he completes his entire revolution in about 
three months. If we watch the course of this planet from night 
to night, we observe it, in its revolution, to cross the ecliptic in 
two opposite points of the heavens, and wander about 7 from 
that plane at its greatest distance from it. Knowing the position 
of the orbit of Mercury with respect to the ecliptic, we may now, 
in imagination, represent that orbit by a great circle passing 
through the center of the planet and the center of the sun, and 
cutting the plane of the ecliptic in two opposite points at an angle 
of 7. We may imagine the intersection of these two great cir- 
cles, with the celestial vault to be marked out in plain and palpable 
lines on the face of the sky ; but we must bear in mind that these 
orbits are mere mathematical planes, having no permanent exist- 
ence in nature, any more than the path of an eagle flying through 
the sky ; and if we conceive of their orbits as marked on the ce- 
lestial vault, we must be careful to attach to the representation 
the same notion as to a thread or wire carried round to trace out 
the course pursued by a horse in a race-ground.* 

The planes of both the ecliptic and the orbit of Mercury, may 
be conceived of as indefinitely extended to a great distance until 
they meet the sphere of the stars ; but the lines which the earth 
and Mercury describe in those planes, that is, their orbits, may be 
conceived of as comparatively near to the sun. Could we now for 
a moment be permitted to imagine that the planes of the ecliptic, 
and of the orbit of Mercury, were made of thin plates of glass, 
and that the paths of the respective planets were marked out on 
their planes in distinct lines, we should perceive the orbit of the 
earth to be almost a perfect circle, while that of Mercury would 

* It would seem superfluous to caution the reader on so plain a point, did not the 
experience of the instructor constantly show that young learners, from the habit of 
seeing the celestial motions represented in orreries and diagrams, almost always fall 
into the absurd notion of considering the orbits of the planets as having a distinct and 
independent existence. 



appear distinctly elliptical. But having once made use of a palpa- 
ble surface and visible lines to aid us in giving position and figure 
to the planetary orbits, let us now throw aside these devices, and 
hereafter conceive of these planes and orbits as they are in nature, 
and learn to refer a body to a mere mathematical plane, and to 
trace its path in that plane through absolute space. 

355. A clear understanding of the motions of Mercury and of 
the relation of its orbit to the plane of the ecliptic, will render it 
easy to understand the same particulars in regard to each of the 
other planets. Standing on the sun we should see each of the 
planets pursuing a similar course to that of Mercury, all moving 
from west to east, with motions differing from each other chiefly 
in two respects, namely, in their velocities, and in the distances 
to which they ever recede from the ecliptic. 

The earth revolves about the sun very much like Venus, and to 
a spectator on the sun, the motions of these two planets would 
exhibit much the same appearances. "We have supposed the ob- 
server to select the plane of the earth's orbit as his standard of 
reference, and to see how each of the other orbits is related to it ; 
but such a selection of the ecliptic is entirely arbitrary ; the spec- 
tator on the sun, who views the motions of the planets as they 
actually exist in nature, would make no such distinction between 
the different orbits, but merely inquire how they were mutually 
related to each other. Taking, however, the ecliptic as the plane 
to which all the others are referred, we do not, as in the case of 
the other planets, inquire how its plane is inclined, nor what are 
its nodes, since it has neither inclination nor node. 

356. Such, in general, are the real motions of the planets, and 
such the appearances which the planetary system would exhibit 
to a spectator at the center of motion. But in order to represent 
correctly the positions of the planetary orbits, at any given time, 
three things must be regarded, the Inclination of the orbit to the 
ecliptic the position of the line of the Nodes and the position of 
the line of the Apsides. In our common diagrams, the orbits are 
incorrectly represented, being all in the same plane, as in the fol- 
lowing diagram, where AEB (Fig. 65,) represents the orbit of 



Fig. 65. 

Mercury as lying in the same plane with the ecliptic. To exhibit 
its position justly (AB being taken as the line of the nodes) it 
should be elevated on one side about 7 and depressed by the 
same number of degrees on the other side, turning on the line 
AB as on a hinge. But even then the representation may be 
incorrect in other respects, for we have taken it for granted that 
the line of the nodes coincides with the line of the apsides, or 
that the orbit of Mercury cuts the ecliptic in the line AB. Whereas, 
it may lie in any given position with respect to the line of the ap- 
sides depending on the longitude of the nodes. If, for example, the 
line of the nodes had chanced to pass through Taurus and Scor- 
pio instead of Cancer and Capricorn, then it would have been repre- 
sented by the line 8 ^l instead of 25V3, and the plane when elevated 
or depressed with respect to the plane of the equator, would be 
turned on this line in our figure.* Moreover, our diagram repre- 
sents the line of the apsides as passing through'Cancer and Cap- 
ricorn, whereas it may have any other position among the signs, 
according to the longitudes of the perigee and apogee. 

* The learner will find it useful to construct such representations of the mutual re 
lations of the planetary orbits of paste board. 


357. The attempt to exhibit the motions of the solar system, and 
the positions of the planetary orbits by means of diagrams, or 
even orreries, is usually a failure. The student who relies exclu- 
sively on such aids as these, will acquire ideas on this subject that 
are both inadequate and erroneous. They may aid reflection, but 
can never supply its place. The impossibility of representing 
things in their just proportions will be evident when we reflect, 
that to do this, if, in an orrery, we make Mercury as large as a 
cherry, we should require to represent the sun by a globe six feet 
in diameter. If we preserve the same proportions in regard to 
distance, we must place Mercury 250 feet, and Uranus 12,500 
feet, or more than two miles from the sun. The mind of the stu- 
dent of astronomy must, therefore, raise itself from such imperfect 
representations of celestial phenomena as are afforded by artificial 
mechanism, and, transferring his contemplations to the celestial 
regions themselves, he must conceive of the sun and planets as 
bodies that bear an insignificant ratio to the immense spaces in 
which they circulate, resembling more a few little birds flying in 
the open sky, than they do the crowded machinery of an orrery. 

358. Having acquired as correct an idea as we are able of the 
planetary system, and of the positions of the orbits with respect to 
the ecliptic, let us next inquire into the nature and causes of the 
apparent motions. 

The apparent motions of the planets are exceedingly unlike the 
real motions, a fact which is owing to two causes ; first, we view 
them out of the center of their orbits ; secondly, we are ourselves in 
motion. From the first cause, the apparent places of the planets 
are greatly changed by perspective ; and from the second cause, 
we attribute to the planets changes of place which arise from our 
own motions of which we are unconscious. 

359. The situation of a heavenly body as seen from the center 
of the sun, is called its heliocentric place ; as seen from the center 
of the earth, its geocentric place. The geocentric motions of the 
planets must, according to what has just been said, be far more 
irregular and complicated than the heliocentric, as will be evident 
from the following diagram, which represents the geocentric mo- 



tions of Mercury for two entire revolutions, embracing a period 
of nearly six months. 

Let S (Fig. 66,) represent the sun, 1, 2, 3, &c. the orbit of Mer- 
cury, a, b, c, &c. that of the earth, and GT the concave sphere of 
the heavens. The orbit of Mercury is divided into 12 equal parts, 
each of which he describes in 7 days, and a portion of the earth's 

Fig. 66. 

orbit described by that body in the time that Mercury describes 
the two complete revolutions, is divided into 24 equal parts. Let 
us now suppose that Mercury is at the point 1 in his orbit, when 
the earth is at the point a ; Mercury will then appear in the heav- 
ens at A. In 7i days Mercury will have reached 2, while the 
earth has reached 6, when Mercury will appear at B. By laying 
a ruler on the point c and 3, d and 4, and so on, in the order of the 


alphabet, the successive apparent places of Mercury in the heavens 
will be obtained. 

From A to C, the apparent motion is direct, or in the order of 
the signs ; from C to G it is retrograde ; at G it is stationary 
awhile, and then direct through the whole arc GT. At T the 
planet is again stationary, and afterwards retrograde along the 
arc TX. 

360. Venus exhibits a variety of motions similar to those of 
Mercury, except that the changes do not succeed each other so 
rapidly, since her period of revolution approaches much more 
nearly to that of the earth. 

361. The apparent motions of the superior planets, are, like 
those of Mercury and Venus, alternately direct, stationary, and 
retrograde. In this case, however, the earth moves faster than 
the planet, and the planet has its opposition but no inferior con- 
junction, whereas an inferior planet has its inferior conjunction, 
but no opposition. These differences render the apparent motions 
of the superior planets in some respects unlike those of Mercury 
and Venus. When a superior planet is in conjunction, its motion 
is direct, because, as in the case of Venus in her superior conjunc- 
tion, (see Fig. 60,) the only effect of the earth's motion is to ac- 
celerate it; but when the planet is in opposition, the earth is 
moving past it with a greater velocity, and makes the planet seem 
to move backwards, like the apparent backward motion of a ves- 
sel when we overtake it and pass rapidly by it in a steamboat. 

362. But the various motions of a superior planet will be best 
understood from a diagram. Hence, let S (Fig. 67,) be the sun ; 
B, C, D, E, the orbit of the earth ; &, c, d, &c. the orbit of a supe- 
rior planet, as Jupiter for example ; and I'E' a portion of the con- 
cave sphere of the heavens. Let bm be the arc described by Ju- 
piter in the time the earth describes the arc BM ; let be, cd, and 
de, &c. be described by Jupiter while the earth describes BC, 
CD, and DE. Now when the earth is at B and Jupiter at b t he 
will appear in the heavens at B'. When the earth reaches C, the 
planet reaches c and will be seen at C', his motions having been 


direct from west to east. While the earth moves from C to D 
and from D to E, Jupiter has moved from c to d, and from d to 
e, and will appear to have advanced among the stars from C' to 
D', and from D' to E', his motion being still direct, but slower 
than before, as he has passed over only the space D'E' in the 
same time that he before moved through the greater spaces B'C' 
and C'D'. 

During the motion of the earth from E to F, and of Jupiter 
from e to/, the earth passes by Jupiter ; and not being conscious 
of our own motion, Jupiter seems to us to have moved backward 
from E' to F'. At E' where the direct motion was changed to a 
retrograde, he would appear to be stationary. Upon the arrival 
of the earth at G, and of Jupiter at g, in opposition to the sun, Ju- 
piter will appear at G', having moved with apparently great ve- 
locity over a large space F'G'. While the earth passes from G to 
H, and from H to I, and Jupiter from g to h, and from h to ', he 


will appear to have moved from G' to P. At P he will again 
appear stationary in the heavens ; but when he advances from i to 
k in the time the earth moves from I to K, he has described the 
arch I'K', and has therefore resumed his direct motion from west 
to east. While the earth moves from K to L and from L to M, 
and Jupiter through the corresponding spaces kl and Zra, the planet 
will appear still to continue his direct motion from K' to L' and 
from L' to M' in the heavens. 

Thus, during a period of six months, while the earth is perform- 
ing one half of her annual circuit, Jupiter has a diversity of mo- 
tions, all performed within a small portion of the heavens. 





363. IN chapter II, we have shown that the figure of the earth's 
orbit is an ellipse, having the sun in one of its foci, and that the 
earth's radius vector describes equal areas in equal times ; and in 
Chapter III, we have remarked that these are only particular 
examples under the law of Universal Gravitation, as is also the 
additional fact, that the squares of the periodical times of the 
planets are as the cubes of the major axes of their orbits. We 
may now learn, more particularly, the process by which the illus- 
trious Kepler was conducted to the discovery of these grand laws 
of the planetary system. 

364. Ptolemy, while he held that the orbits of the planets were 
perfect circles in which the planets revolved uniformly about the 
earth, was nevertheless obliged to suppose that the earth was 
situated out of the center of the circles, and that at the same 

* See Article IV. of the Addenda. 


distance on the other side of the center was situated the point 
(punctum cequans) about which the angular motion of the body 
was equable and uniform. On nearly the same suppositions, Ty- 
cho Brahe had made a great number of very accurate observations 
on the planetary motions, which served Kepler as standards of 
comparison for results, which he deduced from calculations founded 
on the application of geometrical reasoning to hypotheses of his own. 
Kepler first applied himself to investigate the orbit of Mars, the 
motions of which planet appeared more irregular than those of 
any other, except Mercury, which, being seldom seen, had then 
been very little studied. According to the views of Ptolemy and 
Tycho, he at first supposed the orbit to be circular, and the planet 
to move uniformly about a point at a certain distance from the 
sun. He made seventy suppositions before he obtained one that 
agreed with observation, the calculation of which was extremely 
long and tedious, occupying him more than five years.* The sup- 
position of an equable motion in a circle, however varied, could 
not be made to conform to the observations of Tycho, whereas the 
supposition that the orbit was of an oval figure, depressed at the 
sides, but coinciding with a circle at the perihelion, agreed very 
nearly with observation. Such a figure naturally suggested the 
idea of an ellipse, and reasoning on the known properties 
of the ellipse, and comparing the results of calculation with 
actual observation, the agreement was such as to leave no doubt 
that the orbit of Mars is an ellipse, having the sun in one of the 
foci. He immediately conjectured that the same is true of the 
orbits of all the other planets, and a similar comparison of this 
hypothesis with observation, confirmed its truth. Hence he 
established the first great law, that the planets revolve about the 
sun in ellipses, having the sun in one of the foci. 

365. Kepler also discovered from observation, that the velocities 
of the planets when in their apsides, are inversely as their dis- 
tances from the sun, whence it follows that they describe, in these 

* Si te hujus laboriosffl methodi pertsesum fuerit, jure mei te misereat, qui earn ad 
minimum septuagies ivi cum plurima temporis jactura ; et mirari desines hunc quintum 
Jam annum abire, ex quo Martem aggressus sum. 


points, equal areas about the sun in equal times. Although he 
could not prove, from observation, that the same was true in 
every point of the orbit, yet he had no doubt that it was so. 
Therefore, assuming this principle as true, and hence deducing the 
equation of the center, (Art. 200,) he found the result to agree 
with observation, and therefore concluded in general, that the 
planets describe about the sun equal areas in equal times. 

366. Having, in his researches that led to the discovery of the 
first of the above laws, found the relative mean distances of the 
planets from the sun, and knowing their periodic times, Kepler 
next endeavored to ascertain if there was any relation between 
them, having a strong passion for finding analogies in nature. 
He saw that the more distant a planet was from the sun, the 
slower it moved ; so that the periodic times of the more distant 
planets would be increased on two accounts, first, because they 
move over a greater space, and secondly, because their motions in 
their orbits are actually slower than the motions of the planets 
nearer the sun. Saturn, for example, is 9^ times further from the 
sun than the earth is, and the circle described by Saturn is greater 
than that of the earth in the same ratio ; and since the earth re- 
volves around the sun in one year, were their velocities equal, the 
periodic time of Saturn would be 9| years, whereas it is nearly 30 
years. Hence it was evident, that the periodic times of the plan- 
ets increase in a greater ratio than their distances, but in a less 
ratio than the squares of their distances, for on that supposition 
the periodic time of Saturn would be about 90 years. Kepler 
then took the squares of the times and compared them with the 
cubes of the distances, and found an exact agreement between 
them. Thus he discovered the famous law, that the squares of the 
periodic times of all the planets, are as the cubes of their mean dis- 
tances from the sun.* 

This law is strictly true only in relation to planets whose mian- 
thy of matter in comparison with that of the central body is 
inappreciable. When this is not the case, the periodic time is 
shortened in the ratio of the square root of the sun's mass divi- 

* Vince's Complete System, I, 98. 


ded by the sun's plus the planet's mass /- ) . The mass of 

\ M+w/ 

most of the planets is so small compared with the sun's, that this 
modification of the law is unnecessary except where extreme ac- 
curacy is required. 


367. The particulars necessary to be known in order to deter- 
mine the precise situation of a planet at any instant, are called 
the Elements of its Orbit. They are seven in number, of which 
the first two determine the absolute situation of the orbit, and the 
other five relate to the motion of the planet in its orbit. These 
elements are, 

(1.) The position of the line of the nodes. 

(2.) The inclination to the ecliptic. 

(3.) The periodic time. 

(4.) The mean distance from the sun, or semi-axis major. 

(5.) The eccentricity. 

(6.) The place of the perihelion. 

(7.) The place of the planet in its orbit at a particular epoch. 

368. It may at first view be supposed that we can proceed to 
find the elements of the orbit of a planet in the same manner as 
we did those of the solar or lunar orbit, namely, by observations 
on the right ascension and declination of the body, converted into 
latitudes and longitudes by means of spherical trigonometry, (See 
Art. 132.) But in the case of the moon, we are situated in the 
center of her motions, and the apparent coincide with the real 
motions ; and, in respect to the sun, our observations on his appa- 
rent motions give us the earth's real motions, allowing 180 differ- 
ence in longitude. But as we have already seen, the motions of 
the planets appear exceedingly different to us, from what they 
would if seen from the center of their motions. It is necessary 
therefore to deduce from observations made on the earth the cor- 
responding results as they would be if viewed from the center of 
the sun ; that is, in the language of astronomers, having the geo- 
centric place of a planet, it is required to find its heliocentric place. 


3G9. The first steps in this process are the same as in the case of 
the sun and moon. That is, for the purpose of finding the right 
ascension and declination, the planet is observed on the meridian 
with the Transit Instrument and Mural circle, (See Arts. 155 and 
230,) and from these observations, the planet's geocentric longitude 
and latitude are computed by spherical trigonometry. The distance 
of the planet from the sun is known nearly by Kepler's law. From 
these data it is required to find the heliocentric longitude and lati- 

Let S and E (Fig. 68,) be the sun and earth, P the planet, PO 
a line drawn from P perpendicular to the ecliptic, SA the direction 
of Aries, and EH parallel to SA, and therefore (on account of 
the immense distance of the fixed stars) also in the direction of 
Aries. Then OEH, being the apparent distance of the planet 
from Aries in the direction of the ecliptic, is the geocentric longi- 
tude, and OEP, being the apparent distance of the planet from the 
ecliptic taken on a secondary to the ecliptic, is the geocentric 
latitude. It is obvious also that the angles OSA and PSO are 

Fig. 68. 

the heliocentric longitude and latitude. The planet's angular dis- 
tance from the sun, PES, is also known from observation. Hence, 
in the triangle SEP, we know SP and SE and the angle SEP, from 
which we can find PE ; and knowing PE and the angle PEO, we 
can find OE, since OEP is a right angled triangle. Hence in the 
triangle SEO, ES and EO, and the angle SEO (=OEH-SEH= 
difference of longitude of the planet and the sun) are known, and 
hence we can obtain OSE, (Art. 135,) which added to the sun's lon- 
gitude ESA, gives us OSA the planet's heliocentric longitude. 


Also, because PS : Rad. :: OP : Sin. PSO 

A PSxSin. PSO=OPxRad. 
But EP : Rad. : : OP : Sin. OEP 

/. EPxSin. OEP-OPxRad. 

/. PSxSin. PSO=EPxSin. OEP 

/. PS : EP :: Sin. OEP : Sin. PSO. 

The first three terms of this proposition being known, the last 
is found, which is the heliocentric latitude.* 

370. Having now learned how observations made at the earth 
may be converted into corresponding observations made at the 
sun, we may proceed to explain the mode of finding the several 
elements before enumerated ; although our limits will not permit 
us to enter further into detail on this subject, than to explain the 
leading principles on which each of these elements is determined.-]- 

371. First, to determine the position of the Nodes, and the In- 
clination of the Orbit. 

These two elements, which determine the situation of the orbit, 
(Art. 367,) may be derived from two heliocentric longitudes and 
latitudes. Let AR and AS (Fig. 69,) Fig. 69. 

be two heliocentric longitudes, PR and 
QS the heliocentric latitudes, and N 
the ascending node. Then, by Napier's 

theorem, (Art. 132,) Ar ~ R s 

Sin. NR (=AR-AN)_ _sin. NS (=AS- AN) 

tan. PR tan. QS 

Sin. ARxcos. AN cos. ARxsin. ANJ_ 

tan. PR. 
sin. AS x cos. AN cos. AS x sin. AN 

tan. QS~ 

sin. AN_Sin. ARxtan. QS sin. AS x tan. PR 
But tan. AJN cos. AN~C^s. ARxtan. QS-cos. ASxtan. PR* 

* Brinkley's Elements of Astronomy, p. 164. 

t Most of these elements admit of being determined in several different ways, an 
explanation of which may be found in the larger works on Astronomy, as Vince's Com- 
plete System, Vol. 1. Gregory's Ast. p. 212. Woodhouse, p. 562. 

t Day's Trig. Art. 208. 


But AN is the longitude of the ascending node ; and its value 
is found in terms of the heliocentric longitudes and latitudes pre- 
viously determined, (Art. 369.) 

Again, since AN is found, we may deduce from the first equa- 
tion above the value of PNR, which is the inclination of the orbit.* 

372. Secondly, to find the Periodic Time. 

This element is learned, by marking the interval that passes 
from the time when a planet is in one of the nodes until it returns 
to the same node. We may know when a planet is at the node 
because then its latitude is nothing. If, from a series of observa- 
tions on the right ascension and declination of a planet, we deduce 
the latitudes, and find that one of the observations gives the lati- 
tude 0, we infer that the planet was at that moment at the node. 
But if, as commonly happens, no observation gives exactly 0, then 
we take two latitudes that are nearest to 0, but on opposite sides 
of the ecliptic, one south and the other north, and as the sum of the 
arcs of latitude is to the whole interval, so is one of the arcs to the 
corresponding time in which it was described, which time being 
added to the first observation, or subtracted from the second, will 
give the precise moment when the planet was at the node. 

By repeated observations it is found, that the nodes of the planets 
have a very slow retrograde motion. 

373. If the orbit of a planet cut the ecliptic at right angles, then 
small changes of place would be attended by appreciable differ- 
ences of latitude ; but in fact the planetary orbits are in general 
but little inclined to the ecliptic, and some of them He almost in 
the same plane with it. Hence arises a difficulty in ascertaining 
the exact time when a planet reaches its node. Among the most 
valuable observations for determining the elements of a planet's 
orbit, are those made when a superior planet is in or near its oppo- 
sition to the sun, for then the heliocentric and geocentric longitudes 
are the same. When a number of oppositions are observed, the 
planet's motion in longitude as would be observed from the sun will 
be known. The inferior planets also, when in superior conjunction, 

* Brinkley, p. 166. 


have their geocentric and heliocentric longitudes the same. When 
in inferior conjunction, these longitudes differ 180 ; but the in- 
ferior planets can seldom be observed in superior conjunction, on 
account of their proximity to the sun, nor in inferior conjunction 
except in their transits, which occur too rarely to admit of obser- 
vations sufficiently numerous. Therefore, we cannot so readily 
ascertain by simple observation, the motions of the inferior planets 
seen from the sun, as we can those of the superior.* 

Hence, in order to obtain accurately the periodic time of a 
planet, we find the interval elapsed between two oppositions sep- 
arated by a long interval, when the planet was nearly in the same 
part of the Zodiac. From the periodic time, as determined ap- 
proximately by other methods, it may be found when the planet 
has the same heliocentric longitude as at the first observation. 
Hence the time of a complete number of revolutions will be 
known, and thence the time of one revolution. The greater the 
interval of time between the two oppositions, the more accurately 
the periodic time will be obtained, because the errors of observa- 
tion will be divided between a great number of periods ; there- 
fore by using very accurate observations, much precision may be 
attained. For example, the planet Saturn was observed in the 
year 228 B. C. March 2, (according to our reckoning of time,) to 
be near a certain star called j Virginis, and it was at the same 
time nearly in opposition to the sun. The same planet was again 
observed in opposition to the sun, and having nearly the same 
longitude, in Feb. 1714. The exact difference between these dates 
was 1943y. 118d. 21h. 15m. It is known from other sources, that 
the time of a revolution is 29% years nearly, and hence it was 
found that in the above period there were 66 revolutions of Saturn ; 
and dividing the interval by this number, we obtain 29.444 years, 
which is nearly the periodic time of Saturn according to the most 
accurate determination. 

374. Thirdly, to determine the distance from the sun, and major 
axes of the planetary orbits. 

The distance of the earth from the sun being known, the mean 

* Brinkley, p. 167. 


distance of any planet (its periodic time being known) may be 
found by Kepler's law, that the squares of the periodic times are 
as the cubes of the distances. The method of finding the dis- 
tance of an inferior planet from the sun by observations at the 
greatest elongation, has been already explained, (See Art. 308.) 
The distance of a superior planet may be found from observations 
on its retrograde motion at the time of opposition. The periodic 
times of two planets being known, we of course know their mean 
angular velocities, which are inversely as the times. Therefore, 
let Ee (Fig. 70,) be a very small portion of the earth's orbit, and 
Mm a corresponding portion of that of a superior planet, described 
on the day of opposition, about the sun S, on which day the three 
bodies lie in one straight line SEMX. Then the angle ESe and 
MS/w, representing the respective angular velocities of the two 

bodies are known. Now if em be joined, and prolonged to meet 
SM continued in X, the angle EXe, which is equal to the alternate 
angle Xey, being equal to the retrogradation of the planet in the 
same time (being known from observation) is also given. Ee, 
therefore, and the angle EXe being given in the right angled tri- 
angle EXe, the side EX is easily calculated, and thus SX becomes 
known. Consequently, in the triangle SwiX, we have given the 
side SX, and the two angles mSX and wXS, whence the other 
sides Sm and wzX are easily determined. Now Sm is the radius of 
the orbit of the superior planet required, which in this calculation 
is supposed circular as well as that of the earth, a supposition not 
exact, but sufficiently so to afford a satisfactory approximation 
to the dimensions of its orbit, and which, if the process be often 
repeated, in every variety of situation at which the opposition can 
occur, will ultimately afford an average or mean value of its dis- 
lance fully to be depended on.* 

375. The transverse or major axes of the planetary orbits remain 

SirJ. Herschel. 



always the same. Amidst all the perturbations to which other ele- 
ments of the orbit are subject, the line of the apsides is of the same 
invariable length. It is no matter in what direction the planet may 
be moving at that moment. Various circumstances will influence 
the eccentricity and the position of the ellipse, but none of them 
affects its length. 

376. Fourthly, to determine the place of the perihelion the epoch 
of passing the perihelion and the eccentricity. 

There are various methods of finding the eccentricity of a 
planet's orbit and the place of the perihelion, and of course the 
position of the line of the apsides. One is derived from thegreat- 
est equation of the center, (Art. 200.) The greatest equation is the 
greatest difference that occurs between the mean and the true 
motion of a body revolving in an ellipse. It will be necessary 
first to explain the manner in which the greatest equation is found. 

Let AEBF (Fig. 71,) be the orbit of the planet, having the sun 
in the focus at S. In an ellipse, the square root of the product of 
the semi-axes gives the radius of a circle of the same area as the 

Fig. 71. 

ellipse.* Therefore with the center 
S, at the distance SE=VAKxOK, 
describe the circle CEGF, then will 
the area of this circle be equal to that 
of the ellipse. At the same time that 
a planet departs from A the aphelion, 
a body begins to move with a uniform 
motion from C through the periphery 
CEGF, and performs a whole revolu- 
tion in the same period that the planet 
describes the ellipse ; the motion of 
this body will represent the equal or 
mean motion of the earth, and it will 
describe around S areas or sectors 
of circles which are proportional to the times, and equal to the 
elliptic areas described in the same time by the planet. Let the 
equal motion, or the angle about S proportional to the time, be 

* Day's Mensuration. 


CSM, and take ASP equal to the sector CSM ; then the place of 
the planet will be P ; MSC will be the mean anomaly, (Art. 200,) . 
DSC the true anomaly, and MSD the equation of the center. Since 
the sectors CSM and ASP are equal, and the part CSD is common 
to both, PACD and SDM are equal ; and since the areas of circu- 
lar sectors are proportional to their arcs, the equation of the center 
is greatest when the area ACPD is greatest, that is, at the point 
E where the ellipse and circle intersect one another. For when 
the planet descends further, to R for instance, the equation becomes 
proportional to the difference of the areas ACE and wER, or to 
the area GBR/w, V being the situation of the body moving equa- 
bly ; for the sector CSV will be equal to the elliptic area ASR, 
and taking away the common space CERS, then ACE RE?=the 
sector VSwz=the equation. At the points E and F, where the 
circle and ellipse intersect, the radius vector of the planet and the 
radius of the circle of equable motion are equal, and of course those 
radii then describe equal areas in equal times; hence, when the 
real motion of the earth is equal to the mean motion, the equation 
of the center is greatest.*' The mean motion for any given time 
is easily found; for the periodic time : 360:: the given time : the 
number of degrees for that time. Observation shows when the 
actual motion of the planet is the same with this. 

377. Now the equation of the center is greatest twice in the 
revolution, on opposite sides of the orbit, as at E and F, which 
points lie at equal distances from the apsides ; and since the whole 
arc EAF or EBF is known from the time occupied in describing 
it, therefore, by bisecting this arc, we find the points A and B, 
the aphelion and perihelion, and consequently the position of the 
line of the apsides. The time of describing the area EBF being 
known, by bisecting this interval, we obtain the moment of passing 
the perihelion, which gives us the place of the planet in its orbit at 
a particular epoch. 

The amount of the greatest equation obviously depends on the 
eccentricity of the orbit, since it arises wholly from the departure 
of the ellipse from the figure of a perfect circle ; hence, the greatest 

* Gregory's Astronomy, p. 197. 


equation affords the means of determining the eccentricity itself. 
In orbits of small eccentricity, as is the case with most of the 
planetary orbits, it is found that the a.rc which measures the greatest 
equation is very nearly equal to the distance between the foci,* 
which always equals twice the eccentricity, the eccentricity being 
the distance from the center to the focus. Consequently, 57 17' 
44".8f : rad. : : half the greatest equation : the eccentricity. 

The foregoing explanations of the methods of finding the ele- 
ments of the orbits, will serve in general to show the learner how 
these particulars are or may be ascertained ; yet the methods actu- 
ally employed are usually more refined and intricate than these. 
In astronomy scarcely an element is presented simple and unmixed 
with others. Its value when first disengaged, must partake of the 
uncertainty to which the other elements are subject ; and can be 
supposed to be settled to a tolerable degree of correctness, only 
after multiplied observations and many revisions.J 

So arduous has been the task of finding the elements of the 
planetary orbits. 


378. It would seem at first view very improbable, that an in 
habitant of this earth would be able to weigh the sun and planets, 
and estimate the exact quantity of matter which they severally con- 
tain. But the principles of Universal Gravitation conduct us to 
this result, by a process remarkable for its simplicity. By com- 
paring the relations of a few elements that are known to us, we 
ascend to the knowledge of such as appeared beyond the pale of 
human investigation. We learn the quantity of matter in a body 
by the force of gravity it exerts. Let us see how this force is ascer- 

379. The quantities of matter in two bodies, may be found in 
terms of the distances and periodic times of two bodies revolving 
around them respectively, being as the cubes of the distances divided 
by the squares of the periodic times. 

* Vince's Complete System, I, 113. 

t The value of an arc equal to radius ; for 3.14159 : 1 : : 180 : 57o 17' 44".8. 

\ Woodhouse, p. 579. 


The force of gravity G in a body whose quantity of matter is 
M and distance D, varies directly as the quantity of matter, and 

inversely as the square of the distance ; that is, G oc -p. But it 

is shown by writers on Central Forces, that the force of gravity 
also varies as the distance divided by the square of the periodic 

time, or G a. Therefore, a p2, anc ^ M a-^. Thus we may 

find the respective quantities of matter in the earth and the sun, 
by comparing the distance and periodic time of the moon, revolving 
around the earth, with the distance and periodic time of the earth 
revolving around the sun. For the cube of the moon's distance 
from the earth divided by the square of her periodic time, is to the 
cube of the earth's distance from the sun divided by the square of 
her periodic time, as the quantity of matter in the earth is to that 

23S545 3 95,000,000 3 
m the sun. That is, : : : 1 : 353,385. The most 

exact determination of this ratio, gives for the mass of the sun 
354,936 times that of the earth. Hence it appears that the sun 
contains more than three hundred and fifty-four thousand times as 
much matter as the earth. Indeed the sun contains eight hundred 
times as much matter as all the planets. 

Another view may be taken of this subject which leads to the 
same result. Knowing the velocity of the earth in its orbit, we 
may calculate its centrifugal force. Now this force is counter- 
balanced, and the earth retained in its orbit, by the attraction of 
the sun, which is proportional to the quantity of matter in the sun. 
Therefore we have only to see what amount of matter is required 
in order to balance the earth's centrifugal force. It is found that 
the earth itself or a body as heavy as the earth acting at the dis- 
tance of the sun, would be wholly incompetent to produce this 
effect, but that in fact it would take more than three hundred and 
fifty-four thousand such bodies to do it. 

380. The mass of each of the other planets that have satellites 
may be found, by comparing the periodic time of one of its satel- 
lites with its own periodic time around the sun. By this means 
we learn the ratio of its quantity of matter to that of the sun. 


The masses of those planets which have no satellites, as Venus or 
Mars, have been determined, by estimating the force of attraction 
which they exert in disturbing the motions of other bodies. Thus, 
the effect of the moon in raising the tides, leads to a knowledge 
of the quantity of matter in the moon ; and the effect of Venus in 
disturbing the motions of the earth, indicates her quantity of mat- 

381. The quantity of matter in bodies varies as their magnitudes 
and densities conjointly. Hence, their densities vary as their 
masses divided by their magnitudes ; and since we know the mag- 
nitudes of the planets, and can compute as above their masses, we 
can thus learn their densities, which, when reduced to a common 
standard, give us their specific gravities, or show us how much 
heavier they are than water. Worlds therefore are weighed with 
almost as much ease as a pebble, or an article of merchandize. 

The densities and specific gravities of the sun, moon, and planets, 
are estimated as follows :f 

Density. Specific Gravity. 

Sun, .... 0.2543 1.40J 

Moon, .... 0.6150 3.37 

Mercury, .... 2.7820 15.24 

Venus, .... 0.9434 5.17 

Earth, . . . 1.0000 5.48 

Mars, . 0.1293 0.71 

Jupiter, .... 0.2589 1.42 

Saturn, .... 0.1016 0.56 

Uranus 0.2797 1.53 

From this table it appears, that the sun consists of matter but 
little heavier than water ; but that the moon is more than three 
times as heavy as water, though less dense than the earth. It also 
appears that the planets near the sun are, as a general fact, more 

* These estimates are made by the most profound investigations in La Place's Me. 
canique Celeste, Vol. III. 

t Francoeur. 

{ The earth being taken, according to Bailly, at 5.48, the specific gravities of the 
other bodies (which are found by multiplying the density of each by the specific gravity 
of the earth) are here stated somewhat higher than they are given in most works. 


dense than those more remote, Mercury being as heavy as the 
heaviest metals except two or three, while Saturn is as light as a 
cork. The decrease of density however is not entirely regular, 
since Venus is a little lighter than the earth, while Jupiter is 
heavier than Mars, and Uranus than Saturn. 

382. The perturbations occasioned in the motions of the planets 
by their action on each other are very numerous, since every body 
in the system exerts an attraction on every other, in conformity 
with the law of Univ ersal Gravitation. Venus and Mars, approach- 
ing as they do at times comparatively near to the earth, sensibly 
disturb its motions, and the satellites of the remote planets greatly 
disturb each other's movements. 


383. The derangement which the planets produce in the motion 
of one of their number will be very small in the course of one 
revolution ; but this gives us no security that the derangement may 
not become very large in the course of many revolutions. The 
cause acts perpetually, and it has the whole extent of time to work 
in. Is it not easily conceivable then that in the lapse of ages, the 
derangements of the motions of the planets may accumulate, the 
orbits may change their form, and their mutual distances may be 
much increased or diminished? Is it not possible that these 
changes may go on without limit, and end in the complete subver- 
sion and ruin of the system ? If, for instance, the result of this 
mutual gravitation should be to increase considerably the eccen- 
tricity of the earth's orbit, or to make the moon approach contin- 
ally nearer and nearer to the earth at every revolution, it is easy 
to see that in the one case, our year would change its character, 
producing a far greater irregularity in the distribution of the solar 
heat : in the other, our satellite must fall to the earth, occasioning 
a dreadful catastrophe. If the positions of the planetary orbits 
with respect to that of the earth, were to change much, the plan- 
ets might sometimes come very near us, and thus increase the 
effect of their attraction beyond calculable limits. Under such 
circumstances we might have years of unequal length, and seasons 


of capricious temperature ; planets and moons of portentous size 
and aspect glaring and disappearing at uncertain intervals ; tides 
like deluges sweeping over whole continents ; and, perhaps, the 
collision of two of the planets, and the consequent destruction of 
all organization on both of them. The fact really is, that changes 
are taking place in the motions of the heavenly bodies, which have 
gone on progressively from the first dawn of science. The eccen- 
tricity of the earth's orbit has been diminishing from the earliest 
observations to our times. The moon has been moving quicker 
from the time of the first recorded eclipses, and is now in advance 
by about four times her own breadth, of what her own place 
would have been if it had not been affected by this acceleration. 
The obliquity of the ecliptic also, is in a state of diminution, and is 
now about two fifths of a degree less than it was in the time of 

384. But amid so many seeming causes of irregularity, and ruin, 
it is worthy of grateful notice, that effectual provision is made for 
the stability of the solar system. The full confirmation of this fact^ 
is among the grand results of Physical Astronomy. Newton did 
not undertake to demonstrate either the stability or instability of 
the system. The decision of this point required a great number 
of preparatory steps and simplifications, and such progress in the 
invention and improvement of mathematical methods as occu- 
pied the best mathematicians of Europe for the greater part of 
the last century. Towards the end of that time, it was shown by 
La Grange and La Place, that the arrangements of the solar sys- 
tem are stable ; that, in the long run, the orbits and motions remain 
unchanged ; and that the changes in the orbits, which take place 
in shorter periods, never transgress certain very moderate limits. 
Each orbit undergoes deviations on this side and on that side of its 
average state ; but these deviations are never very great, and it 
finally recovers from them, so that the average is preserved. The 
planets produce perpetual perturbations in each other's motions, 
but these perturbations are not indefinitely progressive, but period- 
ical, reaching a maximum value and then diminishing. The pe- 

* Whewell, in the Bridgewater Treatises, p. 128. 


riods which this restoration requires are for the most part enor- 
mous not less than thousands, and in some instances millions of 
years. Indeed some of these apparent derangements, have been 
going on in the same direction from the creation of the world. 
But the restoration is in the sequel as complete as the derange- 
ment ; and in the mean time the disturbance never attains a suf- 
ficient amount seriously to affect the stability of the system.* I 
have succeeded in demonstrating (says La Place) that, whatever be 
the masses of the planets, in consequence of the fact that they all 
move in the same direction, in orbits of small eccentricity, and but 
slightly inclined to each other, their secular irregularities are pe- 
riodical and included within narrow limits ; so that the planetary 
system will only oscillate about a mean state, and will never de- 
viate from it except by a very small quantity. The ellipses of the 
planets have been and always will be nearly circular. The eclip- 
tic will never coincide with the equator ; and the entire extent of 
the variation in its inclination, cannot exceed three degrees. 

385. To these observations of La Place, Professor Whewellf 
adds the following on the importance, to the stability of the solar 
system, of the fact that those planets which have great masses 
have orbits of small eccentricity. The planets Mercury and Mars, 
which have much the largest eccentricity among the old planets, 
are those of which the masses are much the smallest. The mass 
of Jupiter is more than two thousand times that of either of these 
planets. If the orbit of Jupiter were as eccentric as that of Mer- 
cury, all the security for the stability of the system, which analy- 
sis has yet pointed out, would disappear. The earth and the 
smaller planets might in that case change their nearly circular or- 
bits into very long ellipses, and thus might fall into the sun, or fly 
off into remote space. It is further remarkable that in the newly 
discovered planets, of which the orbits are still more eccentric 
than that of Mercury, the masses are still smaller, so that the same 
provision is established in this case also. 

* Whewell, in the Bridgewater Treatises, p. 128. 
t Bridgewater Treatises, p. 131. See also Playfair's Outlines, 2, 290. 




386. A COMET,* when perfectly formed, consists of three parts, 
the Nucleus, the Envelope, and the Tail. The Nucleus, or body 
of the comet, is generally distinguished by its forming a bright 
point in the center of the head, conveying the idea of a solid, or at 
least of a very dense portion of matter. Though it is usually 
exceedingly small when compared with the other parts of the 
comet, yet it sometimes subtends an angle capable of being meas- 
ured by the telescope. The Envelope, (sometimes called the coma,) 
is a dense nebulous covering, which frequently renders the edge 
of the nucleus so indistinct, that it is extremely difficult to ascer- 
tain its diameter with any degree of precision. Many comets have 
no nucleus, but present only a nebulous mass extremely attenuated 
on the confines, but gradually increasing in density towards the 
center. Indeed there is a regular gradation of comets, from such 
as are composed merely of a gaseous or vapory medium, to those 
which have a well defined nucleus. In some instances on record, 
astronomers have- detected with their telescopes small stars through 
the densest part of a comet. 

The Tail is regarded as an expansion or prolongation of the 
coma ; and, presenting as it sometimes does, a train of appalling 
magnitude, and of a pale, portentous light, it confers on this class 
of bodies their peculiar celebrity. 

387. The number of comets belonging to the solar system, is 
probably very great. Many, no doubt, escape observation by being 
above the horizon in the day time. Seneca mentions, that during 
a total eclipse of the sun, which happened 60 years before the 
Christian era, a large and splendid comet suddenly made its 

t K6{ui, coma, from the bearded appearance of comets. 



Fig. 71'. 

Fig. 71". 

COMET OF 1811. COMET OF 1680. 

appearance, being very near the sun. The elements of at least 1 30 
have been computed, and arranged in a table for future compari- 
son. Of these six are particularly remarkable, viz. the comets of 
1680, 1770, and 1811 ; and those which bear the names of Halley, 
Biela, and Encke. The cornet of 1680, was remarkable not only 
for its astonishing size and splendor, and its near approach to the 
sun, but is celebrated for having submitted itself to the observa- 
tions of Sir Isaac Newton, and for having enjoyed the signal honor 
of being the first comet whose elements were determined on the 
sure basis of mathematics. The comet of 1770, is memorable for 
the changes its orbit has undergone by the action of Jupiter, as 
will be more particularly related in the sequel. The comet of 
1811 was the most remarkable in its appearance of all that have 
been seen in the present century. Halley's comet (the same 
which re-appeared in 1835) is distinguished as that whose return 
was first successfully predicted, and whose orbit was first deter- 
mined ; and Biela's and Encke's comets are well known, for their 

236 COMETS. 

short periods of revolution, which subject them frequently to the 
view of astronomers. 

388. In magnitude and brightness comets exhibit a great diver- 
sity. History informs us of comets so bright as to be distinctly 
visible in the day time, even at noon and in the brightest sunshine. 
Such was the comet seen at Rome a little before the assassination 
of Julius Caesar. The comet of 1680 covered an arc of the 
heavens of 97, and its length was estimated at 123,000,000 
miles.* That of 1811, had a nucleus of only 428 miles in diame- 
ter, but a tail 132,000,000 miles long.f Had it been coiled around 
the earth like a serpent, it would have reached round more than 
5,000 times. Other comets are of exceedingly small dimensions, 
the nucleus being estimated at only 25 miles ; and some which are 
destitute of any perceptible nucleus, appear to the largest tele- 
scopes, even when nearest to us, only as a small speck of fog, or 
as a tuft of down. The majority of comets can be seen only by 
the aid of the telescope. 

The same comet, indeed, has often very different aspects, at its 
different returns. Halley's comet in 1305 was described by the 
historians of that age, as cometa horrendce magnitudinis ; in 1456 
its tail reached from the horizon to the zenith, and inspired such 
terror, that by a decree of the Pope of Rome, public prayers were 
offered up at noon-day in all the Catholic churches to deprecate 
the wrath of heaven, while in 1682, its tail was only 30 in length, 
and in 1759 it was visible only to the telescope, until after it had 
passed its perihelion. At its recent return in 1835, the greatest 
length of the tail was about 12.J These changes in the appear- 
ances of the same comet are partly owing to the different positions 
of the earth with respect to them, being sometimes much nearer 
to them when they cross its track than at others ; also one specta 
tor so situated as to see the comet at a higher angle of elevation or 
in a purer sky than another, will see the train longer than it 
appears to another less favorbly situated ; but the extent of the 

* Arago. t Milne's Prize Essay on Comets, 

t But might be seen much longer by indirect vision. (Pro/. Joslin, Am. Jour. Sci- 
ence, 31, 328.) 

COMETS. 237 

changes are such as indicate also a real change in their magnitude 
and brightness. 

389. The periods of comets in their revolutions around the sun, 
are equally various. Encke's comet, which has the shortest known 
period, completes its revolution in 3 years, or more accurately, 
in 1208 days ; while that of 1811 is estimated to have a period of 
3383 years.* 

390. The distances to which different comets recede from the sun, 
are also very various. While Encke's comet performs its entire 
revolution within the orbit of Jupiter, Halley's comet recedes from 
the sun to twice the distance of Uranus, or nearly 3600,000,000 
miles. Some comets, indeed, are thought to go to a much greater 
distance from the sun than this, while some even are supposed to 
pass into parabolic or hyperbolic orbits, and never to return. 

391. Comets shine by reflecting the light of the sun. In one or 
two instances they have exhibited distinct phases,^ although the 
nebulous matter with which the nucleus is surrounded, would com- 
monly prevent such phases from being distinctly visible, even 
when they would otherwise be apparent. Moreover, certain 
qualities of polarized light enable the optician to decide whether 
the light of a given body is direct or reflected ; and M. Arago, of 
Paris, by experiments of this kind on the light of the comet of 
1819, ascertained it to be reflected light. J 

392. The tail of a comet usually increases veiy much as it 
approaches the sun ; and frequently does not reach its maximum 
until after the perihelion passage. In receding from the sun, the 
tail again contracts, and nearly or quite disappears before the body 
of the comet is entirely out of sight. The tail is frequently divi- 
ded into two portions, the central parts, in the direction of the 
axis, being 1 less bright than the marginal parts. In 1744, a comet 
appeared which had six tails, spread out like a fan. 

The tails of comets extend in a direct line from the sun, although 
they are usually more or less curved, like a long quill or feather, 

* Milne. t Delambre, t. 3, p 400. t Francceur, 181. 

238 COMETS. 

being convex on the side next to the direction in which they 
are moving ; a figure which may result from the less velocity of 
the portions most remote from the sun. Expansions of the Enve- 
lope have also been at times observed on the side next the sun,* 
but these seldom attain any considerable length. 

393. The quantity of matter in comets is exceedingly small. 
Their tails consist of matter of such tenuity that the smallest stars 
are visible through them. They can only be regarded as great 
masses of thin vapor, susceptible of being penetrated through 
their whole substance by the sunbeams, and reflecting them alike 
from their interior parts and from their surfaces. It appears, per- 
haps, incredible that so thin a substance should be visible by re- 
flected light, and some astronomers have held that the matter of 
comets is self-luminous ; but it requires but very little light to ren- 
der an object visible in the night, and a light vapor may be visible 
when illuminated throughout an immense stratum, which could not 
,be seen if spread over the face of the sky like a thin cloud. The 
highest clouds that float in our atmosphere, must be looked upon 
as dense and massive bodies, compared with the filmy and all but 
spiritual texture of a comet.f 

394. The small quantity of matter in comets is proved by the 
fact that they have sometimes passed very near to some of the planets 
without disturbing their motions in any appreciable degree. Thus 
the comet of 1770, in its way to the sun, got entangled among the 
satellites of Jupiter, and remained near them four months, yet it did 
not perceptibly change their motions. The same comet also came 
very near the earth ; so near, that, had its mass been equal to that 
of the earth, it would have caused the earth to revolve in an orbit 
so much larger than at present, as to have increased the length of 
the year 2h. 47m. J Yet it produced no sensible effect on the 
length of the year, and therefore its mass, as is shown by La 
Place, could not have exceeded -oVo- f tnat f tne earth, - and 
might have been less than this to any extent. It may indeed be 

* See Dr. Joslin's remarks on Halley's comet, Amer. Jour. Science, Vol. 31. 
t Sir. J. Herschel. \ La Place. 

COMETS. 239 

asked, what proof we have that comets have any matter, and are not 
mere reflexions of light. The answer is that, although they are not 
able by their own force of attraction to disturb the motions of the 
planets, yet they are themselves exceedingly disturbed by the action 
of the planets, and in exact conformity with the laws of universal 
gravitation. A delicate compass may be greatly agitated by the 
vicinity of a mass of iron, while the iron is not sensibly affected 
by the attraction of the needle. 

By approaching very near to a large planet, a comet may have 
its orbit entirely changed. This fact is strikingly exemplified in 
the history of the comet of 1770. At its appearance in 1770, 
its orbit was found to be an ellipse, requiring for a complete revo- 
lution only 5| years ; and the wonder was, that it had not been 
seen before, since it was a very large and bright comet. Astron- 
omers suspected that its path had been changed, and that it had 
been recently compelled to move in this short ellipse, by the dis- 
turbing force of Jupiter and his satellites. The French Institute, 
therefore, offered a high prize for the most complete investigation 
of the elements of this comet, taking into account any circum- 
stances which could possibly have produced an alteration in its 
course. By tracing back the movements of this comet for some 
years previous, to 1770, it was found that, at the beginning of 
1767, it had entered considerably within the sphere of Jupiter's 
attraction. Calculating the amount of this attraction from the 
known proximity of the two bodies, it was found what must have 
been its orbit previous to the time when it became subject to the 
disturbing action of Jupiter. The result showed that it then 
moved in an ellipse of greater extent, having a period of 50 years, 
and having its perihelion instead of its aphelion near Jupiter. It 
was therefore evident why, as long as it continued to circulate in 
an orbit so far from the center of the system, it was never visible 
from the earth. In January, 1767, Jupiter and the comet happened 
to be very near one another, and as both were moving in the same 
direction, and nearly in the same plane, they remained in the 
neighborhood of each other for several months, the planet being 
between the comet and the sun. The consequence was, that the 
comet's orbit was changed into a smaller ellipse, in which its revo- 
lution was accomplished in 5 years. But as it was approaching 

240 COMETS. 

the sun in 1779, it happened again to fall in with Jupiter. It was 
in the month of June, that the attraction of the planet began to 
have a sensible effect ; and it was not until the month of October 
following that they were finally separated. 

At the time of their nearest approach, in August, Jupiter was 
distant from the comet only T ^ T of its distance from the sun, and 
exerted an attraction upon it 225 times greater than that of the 
sun. By reason of this powerful attraction, Jupiter being further 
from the sun than the comet, the latter was drawn out into a new 
orbit, which even at its perihelion came no nearer to the sun than 
the planet Ceres. In this third orbit, the comet requires about 20 
years to accomplish its revolution; and being at so great a dis- 
tance from the earth, it is invisible, and will forever remain so un- 
less, in the course of ages, it may undergo new perturbations, and 
move again in some smaller orbit as before.* 


395. The planets, as we have seen, (with the exception of the 
four new ones, which seem to be an intermediate class of bodies 
between planets and comets,) move in orbits which are nearly cir- 
cular, and all very near to the plane of the ecliptic, and all move 
in the same direction from west to east. But the orbits of comets 
are far more eccentric than those of the planets ; they are in- 
clined to the ecliptic at various angles, being sometimes even 
nearly perpendicular to it ; and the motions of comets are some- 
times retrograde. 

396. The Elements of a comet are five, viz. (1) The perihelion 
distance ; (2) longitude of the perihelion ; (3) longitude of the node ; 
(4) inclination of the orbit ; (5) time of the perihelion passage. 

The investigation of these elements is a problem extremely in- 
tricate, requiring for its solution, a skilful and laborious applica- 
tion of the most refined analysis. Newton himself, pronounced it 
Problema longe difficilimum ; and with all the advantages of the 
most improved state of science, the determination of a comet's 

* Milne. 


orbit is considered one of the most complicated problems in as- 
tronomy. This difficulty arises from several circumstances pecu- 
liar to comets. In the first place, from the elongated form of the 
orbits which these bodies describe, it is during only a very small 
portion of their course, that they are visible from the earth, and 
the observations made in that short period, cannot afterwards be 
verified on more convenient occasions ; whereas in the case of the 
planets, whose orbits are nearly circular, and whose movements may 
be followed uninterruptedly throughout a complete revolution, no 
such impediments to the determination of their orbits occur. There 
is also some unavoidable uncertainty in observations made upon 
bodies whose outlines are so ill-defined. In the second place, there 
are many comets which move in a direction opposite to the order 
of the signs in the zodiac, and sometimes nearly perpendicular to 
the plane of the ecliptic ; so that their apparent course through the 
heavens is rendered extremely complicated, on account of the con- 
trary motion of .he earth. In the third place, as there may be 
a multitude of elliptic orbits, whose perihelion distances are equal, 
it is obvious that, in the case of very eccentric orbits, the slightest 
change in the position of the curve near the vertex, where alone 
the comet can be observed, must occasion a very sensible differ- 
ence in the length of the orbit (as will be obvious from Fig. II" 1 ;) 
and therefore, though a small error produces no perceptible dis- 
crepancy between the observed and the calculated course, while 

Fig. 71"'. 

242 COMETS. 

the comet remains visible from the earth, its effect when diffused 
over the whole extent of the orbit, may acquire a most material or 
even a fatal importance. 

On account of these circumstances, it is found exceedingly diffi- 
cult to lay down the path which a comet actually follows through 
the whole system, and least of all, possible to ascertain with accu- 
racy, the length of the major axis of the ellipse, and consequently 
the periodical revolution.* An error of only a few seconds may 
cause a difference of many hundred years. In this manner, though 
Bessel determined the revolution of the comet of 17G9 to be 2089 
years, it was found that an error of no more than 5" in observation, 
would alter the period either to 2678 years, or to 1692 years. 
Some astronomers, in calculating the orbit of the great comet of 
1680, have found the length of its greater axis 426 times the 
earth's distance from the sun, and consequently its period 8792 
years ; whilst others estimate the greater axis 430 times the earth's 
distance, which alters the period to 8916 years. Newton and 
Halley, however, judged that this comet accomplished its revolu- 
tion in only 570 years. 

397. Disheartened by the difficulty of attaining to any precision 
in that circumstance, by which an elliptic orbit is characterized, 
and, moreover, taking into account the laborious calculations 
necessary for ifs investigation, astronomers usually satisfy them- 
selves with ascertaining the elements of a comet on the supposition 
of its describing a parabola ; and, as this is a curve whose axis is 
infinite, the procedure is greatly simplified by leaving entirely out 
of consideration the periodic revolution. It is true that a parabola 
may not represent with mathematical strictness the course which 
a comet actually follows ; but as a parabola is the intermediate 
curve between the hyperbola and ellipse, it is found that this 
method, which is so much more convenient for computation, also 
accords sufficiently with observations, except in cases when the 
ellipse is a comparatively short one, as that of Encke's comet, for 

* For when we know the length of the major axis, we can find tho periodic time by 
Kepler's law, which applies as well to comets as to planets. 



398. The elements of a comet, with the exception of its periodic 
time, are calculated in a manner similar to those of the planets. 
Three good observations on the right ascension and declination of 
the comet (which are usually found by ascertaining its position 
with respect to certain stars, whose right ascensions and declina- 
tions are accurately known) afford the means of calculating these 

The appearance of the same comet at different periods of its 
return are so various, (Art. 388,) that we can never pronounce a 
given comet to be the same with one that has appeared before, 
from any peculiarities in its physical aspect. The identity of a 
comet with one already on record, is determined by the identity 
of the elements. It was by this means that Halley first established 
the identity of the comet which bears his name, with one that 
had appeared at several preceding ages of the world, of which 
so many particulars were left on record, as to enable him to cal- 
culate the elements at each period. These were as in the follow 
ing table. 

Time of appear. 

Inclin. of the orbit. 

Long, of the Node. 

Long, of Per. 

Per. Dist. 



17 56' 
17 56 
17 02 
17 42 

48 30' 
49 25 
50 21 

50 48 

301 00 
301 39 
302 16 
301 36 



On comparing these elements, no doubt could be entertained 
that they belonged to one and the same body ; and since the in- 
terval between the successive returns was seen to be 75 or 76 
years, Halley ventured to predict that it would again return in 
1758. Accordingly, the astronomers who lived at that period, 
looked for its return with the greatest interest. It was found 
however, that on its way towards the sun it would pass very near 
to Jupiter and Saturn, and by their action on it, it would be re- 
tarded for a long time. Clairaut, a distinguished French mathe- 
matician, undertook the laborious task of estimating the exact 
amount of this retardation, and found it to be no less than 618 
days, namely, 100 days by the action of Jupiter, and 518 days by 
that of Saturn. This would delay its appearance until early in 
the year 1759, and Clairaut fixed its arrival at the perihelion within 

244 COMETS. 

a month of April 13th. It came to the perihelion on the 12th of 

399. The return of Halley's comet in 1835, was looked for with 
no less interest than in 1759. Several of the most accurate math- 
ematicians of the age had calculated its elements with inconceiva- 
ble labor. Their zeal was rewarded by the appearance of the 
expected visitant at the time and place assigned ; it traversed the 
northern sky presenting the very appearances, in most respects, 
that had been anticipated ; and came to its perihelion on the 16th 
of November, within one day of the time predicted by Ponte- 
coulant, a French mathematician who had, it appeared, made the 
most successful calculation.* On its previous return, it was 
deemed an extraordinary achievement to have brought the pre- 
diction within a month of the actual time. 

Many circumstances conspired to render this return of Halley's 
comet an astronomical event of transcendent interest. Of all the 
celestial bodies, its history was the most remarkable ; it afforded 
most triumphant evidence of the truth of the doctrine of univer- 
sal gravitation, and of course of the received laws of astronomy ; 
and it inspired new confidence in the power of that instrument, 
(the Calculus,) by means of which its elements had been investi- 

400. Encke's comet, by its frequent returns, affords peculiar fa- 
cilities for ascertaining the laws of its revolution ; and it has kept 
the appointments made for it, with great exactness. On its re- 
turn in 1839 it exhibited to the telescope a globular mass of 
nebulous matter, resembling fog, and moved towards its perihelion 
with great rapidity. 

But what has made Encke's comet particularly famous, is its 
having first revealed to us the existence of a Resisting Medium in 
the planetary spaces. It has long been a question whether the 
earth and planets revolve in a perfect void, or whether a fluid of 
extreme rarity may not be diffused through space. A perfect 

* See Professor Loomis's Observations on Halley's Comet, Amer. Jour. Science, 30 

COMETS. 245 

vacuum was deemed most probable, because no such effects on the 
motions of the planets could be detected as indicated that they en- 
countered a resisting medium. But a feather or a lock of cotton 
propelled with great velocity, might render obvious the resistance 
of a medium which would not be perceptible in the motions of a 
cannon ball. Accordingly, Encke's comet is thought to have plainly 
suffered a retardation from encountering a resisting medium in the 
planetary regions. The effect of this resistance, from the first dis- 
covery of the comet to the present time, has been to diminish the 
time of its revolution about two days. Such a resistance, by de- 
stroying a part of the projectile force, would cause the comet to 
approach nearer to the sun, and thus to have its periodic time 
shortened. The ultimate effect of this cause will be to bring the 
comet nearer to the sun at every revolution, until it finally falls 
into that luminary, although many thousand years will be required 
to produce this catastrophe.* It is conceivable, indeed, that the 
effects of such a resistance may be counteracted by the attraction 
of one or more of the planets near which it may pass in its succes- 
sive returns to the sun. 

401. It is peculiarly interesting to see a portion of matter of a 
tenuity exceeding the thinnest fog, pursuing its path in space, in 
obedience to the same laws as those which regulate such large and 
heavy bodies as Jupiter or Saturn. In a perfect void, a speck of 
fog if propelled by a suitable projectile force would revolve around 
the sun, and hold on its way through the widest orbit, with as sure 
and steady a pace as the heaviest and largest bodies in the system. 

402. Of the physical nature of comets, little is understood. It is 
usual to account for the variations which their tails undergo by 
referring them to the agencies of heat and cold. The intense heat 
to which they are subject in approaching so near the sun as some 
of them do, is alleged as a sufficient reason for the great expansion 
of thin nebulous atmospheres forming their tails ; and the incon- 
ceivable cold to which they are subject in receding to such a dis- 

* Halley's comet, at its return in 1835, did not appear to be affected by the sup- 
posed resisting medium, and its existence is considered as still doubtful. 

246 COMETS. 

tance from the sun, is supposed to account for the condensation of 
the same matter until it returns to its original dimensions. Thus 
the great comet of 1680 at its perihelion approached 166 times 
nearer the sun than the earth, being only 130,000 miles from the 
surface of the sun.* The heat which it must have received, was 
estimated to be equal to 28,000 times that which the earth receives 
in the same time, and 2000 times hotter than red hot iron. This 
temperature would be sufficient to volatilize the most obdurate 
substances, and to expand the vapor to vast dimensions ; and the 
opposite effects of the extreme cold to which it would be subject 
in the regions remote from the sun, would be adequate to condense 
it into its former volume. 

This explanation however, does not account for the direction 
of the tail, extending as it usually does, only in a line opposite to 
the sun. Some writers therefore, as Delarnbre, suppose that the 
nebulous matter of the comet after being expanded to such a vol- 
ume, that the particles are no longer attracted to the nucleus un- 
less by the slightest conceivable force, are carried off in a direction 
from the sun, by the impulse of the solar rays themselves. f But 
to assign such a power of communicating motion to the sun's rays 
while they have never been proved to have any momentum, is 
unphilosophical ; and we are compelled to place the phenomena 
of comets' tails among the points of astronomy yet to be ex- 

403. Since those comets which have their perihelion very near 
the sun, like the comet of 1680, cross the orbits of all the planets, 
the possibility that one of them may strike the earth, has frequently 
been suggested. Still it may quiet our apprehensions on this sub- 
ject, to reflect on the vast extent of the planetary spaces, in which 
these bodies are not crowded together as we see them erroneously 
represented in orreries and diagrams, but are sparsely scattered at 
immense distances from each other. They are like insects flying 
in the expanse of heaven. If a comet's tail lay with its axis in the 
plane of the ecliptic when it was near the sun, we can imagine that 
the tail might sweep over the earth ; but the tail may be situated 

* See Principia, Lib. in, 41. t Delambre's Astronomy, t. 3, p. 401 

COMETS. 247 

at any angle with the ecliptic as well as in the same plane with it, 
and the chances that it will not be in the same plane, are almost 
infinite. It is also extremely improbable that a comet will cross 
the plane of the ecliptic precisely at the earth's path in that plane, 
since it may as probably cross it at any other point, nearer or 
more remote from the sun. Still some comets have occasionally 
approached near to the earth. Thus Biela's comet in returning to 
the sun in 1832, crossed the ecliptic very near to the earth's track, 
and had the earth been then at that point of its orbit, it might 
have passed through a portion of the nebulous atmosphere of the 
comet. The earth was within a month of reaching that point. 
This might at first view seem to involve some hazard ; yet we must 
consider that a month short implied a distance of nearly 50,000,000 
miles. La Place has assigned the consequences that would ensue 
in case of a direct collision between the earth and a comet ;* but 
terrible as he has represented them on the supposition that the 
nucleus of the comet is a solid body, yet considering a comet (as 
most of them doubtless are) as a mass of exceedingly light nebu- 
lous matter, it is not probable, even were the earth to make its 
way directly through a comet, that a particle of the comet would 
reach the earth. The portions encountered by the earth, would 
be arrested by the atmosphere, and probably inflamed ; and they 
would perhaps exhibit on a more magnificent scale than was ever 
before observed, the phenomena of shooting stars, or meteoric 

* Syst. du Monde, 1. iv, c. 4. 




404. THE FIXED STARS are so called, because, to common ob- 
servation, they always maintain the same situations with respect 
to one another. 

The stars are classed, by their apparent magnitudes. The whole 
number of magnitudes recorded are sixteen, of which the first six 
only are visible to the naked eye ; the rest are telescopic stars. As 
the stars which are now grouped together under one of the first 
six magnitudes are very unequal among themselves, it has recently 
been proposed to subdivide each class into three, making in all 
eighteen instead of six magnitudes visible to the naked eye. 
These magnitudes are not determined by any very definite scale, 
but are merely ranked according to their relative degrees of 
brightness, and this is left in a great measure to the decision of the 
eye alone, although it would appear easy to measure the compar- 
ative degree of light in a star by a photometer, and upon such 
measurement to ground a more scientific classification of the stars. 
The brightest stars to the number of 15 or 20 are considered as 
stars of the first magnitude ; the 50 or 60 next brightest, of the 
second magnitude ; the next 200 of the third magnitude ; and thus 
the number of each class increases rapidly as we descend the scale, 
so that no less than fifteen or twenty thousand are included within 
the first seven magnitudes. 

405. The stars have been grouped in Constellations from the 
most remote antiquity : a few, as Orion, Bootes, and Ursa Major, 
are mentioned in the most ancient writings under the same names 


as they bear at present. The names of the constellations are 
sometimes founded on a supposed resemblance to the objects to 
which the names belong ; as the Swan and the Scorpion were evi- 
dently so denominated from their likeness to those animals ; but 
in most cases it is impossible for us to find any reason for desig- 
nating a constellation by the figure of the animal or the hero which 
is employed to represent it. These representations were probably 
once blended with the fables of pagan mythology. The same fig- 
ures, absurd as they appear, are still retained for the convenience 
of reference ; since it is easy to find any particular star, by speci- 
fying the part of the figure to which it belongs, as when we say a 
star is in the neck of Taurus, in the knee of Hercules, or in the 
tail of the Great Bear. This method furnishes a general clue to 
its position ; but the stars belonging to any constellation are dis- 
tinguished according to their apparent magnitudes as follows : 
first, by the Greek letters, Alpha, Beta, Gamma, &c. Thus a 
Orionis, denotes the largest star in Orion, ft Andromeda, the 
second star in Andromeda, and y Leonis, the third brightest star in 
the Lion. Where the number of the Greek letters is insufficient 
to include all the stars in a constellation, recourse is had to the 
letters of the Roman alphabet, a, b, c, &c. ; and, in cases where 
these are exhausted, the final resort is to numbers. This is evi- 
dently necessary, since the largest constellations contain many 
hundreds or even thousands of stars. Catalogues of particular 
stars have also been published by different astronomers, each 
author numbering the individual stars embraced in his list, accord- 
ing to the places they respectively occupy in the catalogue. 
These references to particular catalogues are sometimes entered 
on large celestial globes. Thus we meet with a star marked 84 
H., meaning that this is its number in HerscheFs catalogue, or 
140 M. denoting the place the star occupies in the catalogue of 

406. The earliest catalogue of the stars was made by Hippar- 
chus of the Alexandrian School, about 140 years before the 
Christian era. A new star appearing in the firmament, he was 
induced to count the stars and to record their positions, in order 
that posterity might be able to judge of the permanency of the con- 



stellations. His catalogue contains all that were conspicuous to 
the naked eye in the latitude of Alexandria, being 1022. Most per- 
sons unacquainted with the actual number of the stars which com- 
pose the visible firmament, would suppose it to be much greater than 
this ; but it is found that the catalogue of Hipparchus embraces 
nearly all that can now be seen in the same latitude, and that on 
the equator, when the spectator has the northern and southern 
hemispheres both in view, the number of stars that can be counted 
does not exceed 3000. A careless view of the firmament in a 
clear night, gives us the impression of an infinite multitude of stars ; 
but when we begin to count them, they appear much more 
sparsely distributed than we supposed, and large portions of the 
sky appear almost destitute of stars. 

By the aid of the telescope, new fields of stars present them- 
selves of boundless extent ; the number continually augmenting 
as the powers of the telescope are increased. Lalande, in his 
Histoire Celeste, has registered the positions of no less than 
50,000; and the whole number visible .in the largest telescopes 
amount to many millions. 

407. It is strongly recommended to the learner to acquaint 
himself with the leading constellations at least, and with a few 
of the most remarkable individual stars. The task of learning 
them is comparatively easy, and hardly any kind of knowledge, 
attained with so little Jabor, so amply rewards the possessor. It 
will generally be advisable, at the outset, to get some one already 
acquainted with the stars, to point out a few of the most conspicu- 
ous constellations, those of the Zodiac for example : the learner 
may then resort to a celestial globe,* and fill up the outline by 
tracing out the principal stars in each constellation as there laid 
down. By adding one new constellation to his list every night, 
and reviewing those already acquired, he will soon become fa- 
miliar with the stars, and will greatly augment his interest and 
improve his intelligence in celestial observation and practical as- 

* For the method of rectifying the globe so as to represent the appearance of the 
heavens on any particular eve'ning, see page 27, Prob. 76. 



408. We will point out particular marks by which the leading 
constellations maybe recognized, leaving it to^ the learner, after 
he has found a constellation, to trace out additional members of 
it by the aid of the celestial globe, or by maps of the stars. Let 
us begin with the Constellations of the Zodiac, which succeeding 
each other as they do in a known order, are most easily found. 

ARIES (The RAM) is a small constellation, known by two bright 
stars which form his head, a and (3 Arietis. These two stars are 
four degrees* apart ; and directly south of ft at the distance of one 
degree, is a smaller star, y Arietis. It has been already inti- 
mated (Art. 193,) that the vernal equinox probably was near the 
head of Aries, when the signs of the Zodiac received the present 

TAURUS (The BULL) will be readily found by the seven stars or 
Pleiades, which lie in his neck. The largest star in Taurus is 
Aldebaran, in the Bull's eye, a star of the first magnitude, of a 
reddish color somewhat resembling the planet Mars. Aldebaran 
and four other stars in the face of Taurus, compose the Hyades. 

GEMINI (The TWINS) is known by two very bright stars, Castor 
and Pollux, five degrees asunder. Castor (the northern) is of the 
first, and Pollux of the second magnitude. 

CANCER (The CRAB). There are no large stars in this constel- 
lation, and it is regarded as less remarkable than any other in the 
Zodiac. It contains however an interesting group of small stars, 
called Prcesepe or the Nebula of Cancer, which resembles a comet, 
and is often mistaken for one, by persons unacquainted with 
the stars. With a telescope of very moderate powers this nebula 
is converted into a beautiful assemblage of exceedingly bright stars. 

LEO (The LION) is a very large constellation, and has many 
interesting members. Regulus (a Leonis) is a star of the first 
magnitude, which lies directly in the ecliptic, and is much used in 
astronomical observations. North of Regulus lies a semi-circle of 
bright stars, forming a sickle of which Regulus is the handle. 

* These measures are not intended to be stated with exactness, but only with such 
a degree of accuracy as may serve for a general guide. 


Denebola, a star of the second magnitude, is in the Lion's tail, 25 
northeast of Regulus. 

VIRGO (The VIRGIN) extends a considerable way from west 
to east,..but contains only a few bright stars. Spica, however, is 
a star of the first s magnitude, and lies a little east of the place of 
the autumnal equinox. Eighteen degrees eastward of Denebola, 
and twenty degrees north of Spica, is Vindemiatrix, in the arm of 
Virgo, a star of the third magnitude. 

LIBRA (The BALANCE) is distinguished by three large stars, of 
which the two brightest constitute the beam of the balance, and 
the smallest forms the top or handle. 

SCORPIO (The SCORPION) is one of the finest of the constella- 
tions. His head is formed of five bright stars arranged in the 
arc of a circle, which is crossed in the center by the ecliptic nearly 
at right angles, near the brightest of the five, fS Scorpionis. Nine 
degrees southeast of this, is a remarkable star of the first magni- 
tude, of a reddish color, called Cor Scorpionis. or Antares. South 
of this a succession of bright stars sweep round towards the east, 
terminating in several small stars, forming the tail of the Scorpion. 

SAGITTARIUS (The ARCHER). Northeast of the tail of the Scor- 
pion, are three stars in the arc of a circle which constitute the bow 
of the Archer, the central star being the brightest, directly west 
of which is a bright star which forms the arrow. 

CAPRICORNUS (The GOAT) lies northeast of Sagittarius, and is 
known by two bright stars, three degrees apart, which form the 

AQUARIUS (The WATER BEARER) is recognized by two stars 
in a line with a Capricorn^ forming the shoulders of the figure. 
These two stars are 10 apart, and 3 southeast is a third star, 
which together with the other two, makes an acute triangle, of 
which the westernmost is the vertex. 

PISCES (The FISHES) lie between Aquarius and Aries. They 
are not distinguished by any large stars, but are connected by a 
series of small stars, that form a crooked line between them. 
Piscis Australis, the Southern Fish, lies directly below Aquarius, 
and is known by a singfe bright star far in the south, having a 
declination of 30. The name of this star is Fomalhaut, and is 
much used in astronomical measurements. 


409. The Constellations of the Zodiac, being first well learned, 
so as to be readily recognized, will facilitate the learning of others 
that lie north and south of them. Let us therefore next review 
the principal Northern Constellations, beginning north of Aries and 
proceeding from west to east. 

ANDROMEDA, is characterized by three stars of the second mag- 
nitude, situated in a straight line, extending from west to east. 
The middle star is about 17 north of /3 Arietis. It is in the girdle 
of Andromeda, and is named Mirach. The other two lie at about 
equal distances, 14 C west and east of Mirach. The western star, 
in the head of Andromeda, lies in the Equinoctial Colure. The 
eastern star, Almaak, is situated in the foot. 

PERSEUS lies directly north of the Pleiades, and contains sev- 
eral bright stars. About 18 from the Pleiades is Algol, a star 
of the second magnitude, in the Head of Medusa, which forms a 
part of the figure ; and 9 northeast of Algol is Algenib, of the 
same magnitude in the back of Perseus. Between Algenib and 
the Pleiades are three bright stars, at nearly equal intervals, which 
compose the right leg of Perseus. 

AURIGA (the WAGONER) lies directly east of Perseus, and extends 
nearly parallel to that constellation from north to south. Capella, 
a very white and beautiful star of the first magnitude, distinguishes 
this constellation. The feet of Auriga are near the Bull's Horns. 

The LYNX comes next, but presents nothing particularly inter- 
esting, containing no stars above the fourth magnitude. 

LEO MINOR consists of a collection of small stars north of the 
sickle in Leo, and south of the Great Bear. Its largest star is 
only of the third magnitude. 

COMA BERENICES is a cluster of small stars, north of Denebola, 
in the tail of the lion, and of the head of Virgo. About 12 
directly north of Berenice's Hair, is a single bright star called Cor 
Caroli, or Charles's Heart. 

BOOTES, which comes next, is easily found by means of Arc- 
turus, a star of the first magnitude, of a reddish color, which is 
situated near the knee of the figure. Arcturus is accompanied 
by three small stars forming a triangle a little to the southwest. 
Two bright stars y and 5 Bootis, form the shoulders, and /3 of the 
third magnitude is in the head of the figure. 


CORONA BOREALIS (The CROWN) which is situated E. of Bootes, 
is very easily recognized, composed as it is of a semi-circle of 
bright stars. In the center of the bright crown, is a star of the 
second magnitude, called gemma ; the remaining stars are all much 

HERCULES, lying between the Crown on the west and the Lyre 
on the east, is very thickly set with stars, most of which are quite 
small. This Constellation covers a great extent of the sky, es- 
pecially from N. to S., the head terminating within 15 of the 
equator, and marked by a star of the third magnitude, called Ras- 
algethi, which is the largest in the Constellation. 

OPHIUCHUS is situated directly south of Hercules, extending some 
distance on both sides of the equator, the feet resting on the Scor- 
pion. The head terminates near the head of Hercules, and like 
that, is marked by a bright star within 5 of a Herculis. Ophiu- 
chus is represented as holding in his hands the SERPENT, the head 
of which, consisting of .three bright stars, is situated a little south 
of the Crown. The folds of the serpent will be easily followed 
by a succession of bright stars which extend a great way to the 

AQUILA (The EAGLE) is conspicuous for three bright stars in its 
neck, of which the central one, Altair, is a very brilliant white 
star of the first magnitude. Antinous lies directly south of the 
Eagle, and north of the head of Capricornus. 

DELPHINUS (The DOLPHIN) is a small but beautiful Constellation, 
a few degrees east of the Eagle, and is characterized by four bright 
stars near to one, another, forming a small rhombic square. An- 
other star of the same magnitude 5 south, makes the tail. 

PEGASUS lies between Aquarius on the southwest and Andromeda 
on the northeast. It contains but few large stars. A very regu- 
lar square of bright stars is composed of a Andromedce, and the 
three largest stars in Pegasus, namely, Scheat, Markab, and Alge- 
nib. The sides composing this square are each about 15. Alge- 
nib is situated in the equinoctial colure. 

410. We may now review the Constellations which surround 

the North Pole, within the circle of perpetual apparition. (Art. 54.) 

URSA MINOR (The LITTLE BEAR) lies nearest the pole. The 


Pole-star, Polaris, is in the extremity of the tail, and is of the third 
magnitude. Three stars in a straight line 4 or 5 apart, com- 
mencing with the Pole-star, lead to a trapezium of four stars, and 
the whole seven form together a dipper, the trapezium being the 
body, and the three stars the handle. 

URSA MAJOR (The GREAT BEAR) is situated between the pole 
and the Lesser Lion, and is usually recognized by the figure of a 
larger and more perfect dipper, which constitutes the hinder part 
of the animal. This has also seven stars, four in the body of the 
dipper, and three in the handle. All these are stars of much ce- 
lebrity. The two in the western side of the dipper, a and j3, are 
called Pointers, on account of their always being in a right line 
with the Pole-star, and therefore affording an easy mode of finding 
that. The first star in the tail, next the body, is named Alioth, and 
the second Mizar. The head of the Great Bear lies far to the 
westward of the Pointers, and is composed of numerous small 
stars ; and the feet are severally composed of two small stars very 
near to each other. 

DRACO (The DRAGON) winds round between the Great and Lit- 
tle Bear ; and commencing with the tail, between the Pointers and 
the Pole-star, it is easily traced by a succession of bright stars ex- 
tending from west to east ; passing under Ursa Minor, it returns 
westward, and terminates in a triangle which forms the head of 
Draco, near the feet of Hercules, northwest of Lyra. 

CEPHEUS lies eastward of the breast of the Dragon, but has no 
stars above the third magnitude. 

^CASSIOPEIA is known by the figure of a chair, composed of four 
stars which form the legs, and two which form the back. This 
Constellation lies between Perseus and Cepheus, in the Milky. 

CYGNUS (The SWAN) is situated also in the Milky Way, some 
distance southwest of Cassiopeia, towards the Eagle. Three 
bright stars, which lie along the Milky Way, form the body and 
neck of the Swan, and two others in a line with the middle one of 
the three, one above and one below, constitute the wings. This 
Constellation is among the few that exhibit some resemblance to 
the animals whose names they bear. 

LYRA (The LYRE) is directly west of the Swan, and is easily 


distinguished by a beautiful white star of the first magnitude, a 

411. The Southern Constellations are comparatively few in 
number. We shall notice only the Whale, Orion, the Greater and 
Lesser Dog, Hydra, and the Crow. 

CETUS (The WHALE) is distinguished rather for its extent than its 
brilliancy, reaching as it does through 40 of longitude, while none 
of its stars except one, are above the third magnitude. Menkar (a 
Ceti) in the mouth, is a star of the second magnitude, and several 
other bright stars directly south of Aries, make the head and neck 
of the Whale. Mira (o Ceti) in the neck of the Whale, is a varia- 
ble star. 

ORION is one of the largest and most beautiful of the constella- 
tions, lying southeast of Taurus. A cluster of small stars form the 
head ; two large stars, Betalgeus of the first and Bellatrix of the 
second magnitude, make the shoulders ; three more bright stars 
compose the buckler, and three the sword ; and Rigel, another 
star of the first magnitude, makes one of the feet. In this Con- 
stellation there are 70 stars plainly visible to the naked eye, inclu- 
ding two of the first magnitude, four of the second, and three of 
the third. 

CANIS MAJOR lies S. E. of Orion, and is distinguished chiefly by 
its containing the largest of the fixed stars, Sirius. 

CANIS MINOR, a little north of the equator, between Canis Major 
and Gemini, is a small Constellation, consisting chiefly of two 
stars, of which Procyon is of the first magnitude. 

HYDRA has its head near Procyon, consisting of a number of 
stars of ordinary brightness. About 15 S. E. of the head, is a 
star of the second magnitude, forming the heart, (Cor Hydras) ; 
and eastward of this, is a long succession of stars of the fourth and 
fifth magnitudes composing the body and the tail, and reaching a 
few degrees south of Spica Virginis. 

CORVUS (The CROW) is represented as standing on the tail of 
Hydra. It consists of small stars, only three of which are as 
large as the third magnitude. 

412. The foregoing brief sketch is designed merely to aid the 


student in finding the principal constellations and the largest fixed 
stars. When we have once learned to recognize a constellation 
by some characteristic marks, we may afterwards fill up the out- 
line by the aid of a celestial globe or a map of the stars. It will be 
of little avail, however, merely to commit this sketch to memory ; 
but it will be very useful for the student at once to render himself 
familiar with it, from the actual specimens which every clear 
evening presents to his view. 




413. IN various parts of the firmament are seen large groups or 
clusters, which, either by the naked eye, or by the aid of the small- 
est telescope, are perceived to consist of a great number of small 
stars. Such are the Pleiades, Coma Berenices, and Prsesepe or 
the Bee-hive, in Cancer. The Pleiades, or Seven Stars, as they 
are called, in the neck of Taurus, is the most conspicuous cluster. 
When we look directly at this group, we cannot distinguish 
more than six stars, but by turning the eye sideways* upon it, we 
discover that there are many more. Telescopes show 50 or 60 
stars crowded together and apparently insulated from the other 
parts of the heavens.f Coma Berenices has fewer stars, but they 
are of a larger class than those which compose the Pleiades. The 
Bee-hive or Nebula of Cancer as it is called, is one of the finest 
objects of this kind for a small telescope, being by its aid con- 
verted into a rich congeries of shining points. The head of Orion 
affords an example of another cluster, though less remarkable than 
the others. 

* Indirect vision is far more delicate than direct. Thus we can see the Zodiacal 
Light or a Comet's Tail, much more distinctly and better defined, if we fix one eye on 
a part of the heavens at some distance, and turn the other eye obliquely upon the ob- 
ject. tSirJ.Herschel. 



414. Nebula are those faint misty appearances which resemble 
comets, or a small speck of fog. The Galaxy or Milky Way pre- 
sents a continued succession of large nebulae. A very remarkable 
Nebula, visible to the naked eye, is seen in the girdle of Androme- 
da. No powers of the telescope have been able to resolve this 
into separate stars. Its dimensions are astonishingly great. In 
diameter it is about 15'. The telescope reveals to us innumerable 
objects of this kind. Sir William Herschel has given catalogues 
of 2000 Nebulae, and has shown that the nebulous matter is dis- 
tributed through the immensity of space in quantities inconceiva- 
bly great, and in separate parcels of all shapes and sizes, and of 
all degrees of brightness between a mere milky appearance and 
the condensed light of a fixed star. Finding that the gradations 
between the two extremes were tolerably regular, he thought it 
probable that the nebulae form the materials out of which nature 
elaborates suns and systems ; and he conceived that, in virtue of 
a central gravitation, each parcel of nebulous matter becomes 
more and more condensed, and assumes a rounder form. He in- 
fers from the eccentricity of its shape, and the effects of the mutual 
gravitation* of its particles, that it acquires gradually a rotary mo- 
tion ; that the condensation goes on increasing until the mass- 
acquires consistency and solidity, and all the other characters of a 
comet or a planet ; that by a still further process of condensation, 
the body becomes a real star, self-shining ; and that thus the waste 
of the celestial bodies, by the perpetual diffusion of their light, is 
continually compensated and restored by new formations of such 
bodies, to replenish forever the universe with planets and stars.* 

415. These opinions are recited here rather out of respect to 
their notoriety and celebrity, than because we suppose them to be 
founded on any better evidence than conjecture. The Philosophi- 
cal Transactions for many years, both before and after the com- 
mencement of the present century, abound with both the ob- 
servations and speculations of Sir William Herschel. The for- 
mer are deserving of all praise ; the latter of less confidence. 
Changes, however, are going on in some of the nebulae, which 

* Phil, Trans. 1811. 


plainly show that they are not, like planets and stars, fixed and 
permanent creations. Thus the great nebula in the girdle of An- 
dromeda, has very much altered its structure since it first became 
an object of telescopic observation.* Many of the nebulae are of 
a globular form, (Fig. 72, a) but frequently they present the ap- 

(Fig. 72, a.) (Fig.72, 6.) 

pearance of a rapid increase of numbers towards the center, (Fig. 
72, b) the exterior boundary being irregular, and the central parts 
more nearly spherical. 

416. The Nebula in the sword of Orion is particularly celebrated, 
being very large and of a peculiarly interesting appearance.f Ac- 
cording to Sir John Herschel, its nebulous character is very dif- 
ferent from what might be supposed to arise from the assemblage 
of an immense collection of small stars. It is formed of little 
flocculent masses like wisps of clouds ; and such wisps seem to 
adhere to many small stars at its outskirts, and especially to one 
considerable star which it envelops with a nebulous atmosphere of 
considerable extent and singular figure. 

Descriptions, however, can convey but a very imperfect idea 
of this wonderful class of astronomical objects, and we would 
therefore urge the learner studiously to avail himself of the first 
opportunity he may have to view them through a large telescope, 
especially the Nebula of Andromeda and of Orion. 

Nebulous Stars are such as exhibit a sharp and brilliant 
star surrounded by a disk or atmosphere of nebulous matter. 
These atmospheres in some cases present a circular, in others an 
oval figure ; and in some instances, the nebula consists of a long, 
narrow spindle-shaped ray, tapering away at both ends to points^ 

Astron. Trans. II, 495. t See Article V. of the Addenda. 


Annular Nebulce also exist, but are among the rarest objects in 
the heavens. The most conspicuous of this class, is to be found 
exactly half way between the stars (3 and 7 Lyrae, and may be seen 
with a telescope of moderate power.* 

Planetary Nebula constitute another variety, and are very re- 
markable objects. They have, as their name imports, exactly the 
appearance of planets. Whatever may be their nature, they must 
be of enormous magnitude. One of them is to be found in the 
parallel of v Aquarii, and about 5m. preceding that star. Its appa- 
rent diameter is about 20". Another in the Constellation An- 
dromeda, presents a visible disk of 12", perfectly defined and 
round. Granting these objects to be equally distant from us with 
the stars, their real dimensions must be such as, on the lowest 
computation, would fill the orbit of Uranus. It is no less evident 
that, if they be solid bodies, of a solar nature, the intrinsic splendor 
of their surfaces must be almost infinitely inferior to that of the 
sun. A circular portion of the sun's disk, subtending an angle of 
20", would give a light equal to 100 full moons; while the objects 
in question are hardly, if at all, discernible with the naked eye.f 

418. The Galaxy or Milky Way is itself supposed by some to 
be a nebula of which the sun forms a component part ; and hence 
it appears so much greater than other nebulae only in consequence 
of our situation with respect to it, and its greater proximity to our 
system. So crowded are the stars in some parts of this zone, that 
Sir William Herschel, by counting the stars in a single field of his 
telescope, estimated that 50,000 had passed under his review in a 
zone two degrees in breadth during a single hour's observation. 
Notwithstanding the apparent contiguity of the stars which crowd 
the galaxy, it is certain that their mutual distances must be incon- 
ceivably great. 

419. VARIABLE STARS are those which undergo a periodical 

* A list of 288 bright nebulas, with references to well known stars, near which they 
are situated, is given in the Edinburg Enyclopaedia, Art. Astronomy, p. 781. It is con- 
venient for finding any required nebula, 

t Sir J. Herschel. 


change of brightness. One of the most remarkable is the star 
Mira in the Whale, (o Ceti.) It appears once in 1 1 months, re- 
mains at its greatest brightness about a fortnight, being then, on 
some occasions, equal to a star of the second magnitude. It then 
decreases about three months, until it becomes completely invisible, 
and remains so about five months, when it again becomes visible, 
and continues increasing during the remaining three months of its 

Another very remarkable variable star is Algol (/3 Persei.) It 
is usually visible as a star of the second magnitude, and continues 
such for 2d. 14h. when it suddenly begins to diminish in splendor, 
and in about 3i hours is reduced to the fourth magnitude. It then 
begins again to increase, and in 3? hours more, is restored to its 
usual brightness, going through all its changes in less than three 
days. This remarkable law of variation appears strongly to sug- 
gest the revolution round it of some opake body, which, when in- 
terposed between us and Algol, cuts off a large portion of its light. 
It is (says Sir J. Herschel) an indication of a high degree of ac- 
tivity in regions where, but for such evidences, we might conclude 
all lifeless. Our sun requires almost nine times this period to per- 
form a revolution on its axis. On the other hand, the periodic 
time of an opake revolving body, sufficiently large, which would 
produce a similar temporary obscuration of the sun, seen from a 
fixed star, would be less than fourteen hours. 

The duration of these periods is extremely various. While that 
of ]8 Persei above mentioned, is less than three days, others are 
more than a year, and others many years. 

420. TEMPORARY STARS are new stars which have appeared 
suddenly in the firmament, and after a certain interval, as suddenly 
disappeared and returned no more. 

It was the appearance of a new star of this kind 125 years be- 
fore the Christian era, that prompted Hipparchus to draw up a 
catalogue of the stars, the first on record. Such also was the star 
which suddenly shone out A. D. 389, in the Eagle, as bright as 
Venus, and after remaining three weeks, disappeared entirely. 
At other periods, at distant intervals, similar phenomena have pre- 
sented themselves. Thus the appearance of a star in 1572, was 


so sudden, that Tycho Brahe returning home one day was sur- 
prized to find a collection of country people gazing at a star, which 
he was sure did not exist half an hour before. It was then as 
bright as Sirius, and continued to increase until it surpassed Jupi- 
ter when brightest, and was visible at mid-day. In a month it 
began to diminish, and in three months afterwards it had entirely 

It has been supposed by some that in a few instances, the same 
star has returned, constituting one of the periodical or variable 
stars of a long period. 

Moreover, on a careful re-examination of the heavens, and a 
comparison of catalogues, many stars are now found to be miss- 

421. DOUBLE STARS are those which appear single to the naked 
eye, but are resolved into two by the telescope ; or if not visible 
to the naked eye, are seen in the telescope very close together. 
Sometimes three or more stars are found in this near connexion, 
constituting triple or multiple stars. Castor, for example, when 
seen by the naked eye, appears as a single star, but in a telescope 
even of moderate powers, it is resolved into two stars of between 
the third and fourth magnitudes, within 5" of each other. These 
two stars are nearly of equal size, but frequently one is exceedingly 
small in comparison with the other, resembling a satellite near its 
primary, although in distance, in light, and in other characteristics, 
each has all the attributes of a star, and the combination therefore 
cannot be that of a planet with a satellite. In some instances, also, 
the distance between these objects is much less than 5", and in 
many cases it is less than 1". The extreme closeness, together 
with the exceeding minuteness of most of the double stars, requires 
the best telescope united with the most acute powers of observa- 
tion. Indeed, certain of these objects are regarded as the severest 
tests both of the excellence of the instrument, and of the skill of the 

422. Many of the double stars exhibit the curious and beautiful 

*SirJ. Herschel. 


phenomena of contrasted or complementary colors.* In such in- 
stances, the larger star is usually of a ruddy or orange hue, while 
the smaller one appears blue or green, probably in virtue of that 
general law of optics, which provides that when the retina is ex- 
cited by any bright colored light, feebler lights which seen alone 
would produce no sensation but of whiteness, appear colored, with 
the tint complementary to that of the brighter. Thus a yellow 
color predominating in the light of the brighter star, that of the 
less bright one in the same field of view will appear blue ; while, 
if the tint of the brighter star verges to crimson, that of the other 
will exhibit a tendency to green, or even under favorable circum- 
stances, will appear as a vivid green. The former contrast is 
beautifully exhibited by Cancri, the latter by y Andromedae, both 
fine double stars. If, however, the colored star is much the 
less bright of the two, it will not materially affect the other. Thus 
for instance, t\ Cassiopeiae exhibits the beautiful combination of a 
large white star, and a small one of a rich ruddy purple. It is by 
no means, however, intended to say, that in all such cases, one of 
the colors is the mere effect of contrast, and it may be easier sug- 
gested in words, than conceived in imagination, what variety of 
illumination two suns, a red and green, or a yellow and a blue sun, 
must afford a planet circulating about either ; and what charming 
contrasts and " grateful vicissitudes," a red and green day for in- 
stance, alternating with a white one and with darkness, might arise 
from the presence or absence of one or other, or both above the 
horizon. Insulated stars of a red color, almost as deep as that of 
blood, occur in many parts of the heavens, but no green or blue star, 
of any decided hue, has, we believe, ever been noticed, unassociated 
with a companion brighter than itself.t 

423. Our knowledge of the double stars almost commenced with 
Sir William Herschel, about the year 1780. At the time he 

* Complementary colors are such as together make white light. If all the colors of 
the spectrum be laid down on a circular ring, each occupying its proportionate space, any 
two colors on the opposite sides of the zone, are complementary to each other, and 
when of the same degree of intensity, they compose white light, firewater's Optics, 
Part in, c. 26. 

t Sir J. HerscheL 


began his search for them, he was acquainted with only four. 
Within five years, he discovered nearly 700 double stars.* In his 
memoirs published in the Philosophical Transactions^ he gave 
most accurate measurements of the distances between the two 
stars, and of the angle which a line joining the two, formed with 
the parallel of declination.J These data would enable him, or at 
least posterity, to judge whether these minute bodies ever change 
their position with respect to each other. 

Since 1821, these researches have been prosecuted with great 
zeal and industry by Sir James South and Sir John Herschel in 
England, and by Professor Struve at Dorpat in Russia, and the 
whole number of double stars now known, amounts to several 
thousands. Two circumstances add a high degree of interest 
to the phenomena of the double stars, the first is, that a few of 
them at least are found to have a revolution around each other, and 
the second, that they are supposed to afford the means of obtain- 
ing the parallax of the fixed stars. Of these topics we shall treat 
in the next chapter. 



424. IN 1803, Sir William Herschel first determined and an- 
nounced to the world, that there exist among the stars, separate 
systems, composed of two stars revolving about each other in regu- 
lar orbits. These he denominated Binary Stars, to distinguish 
them from other double stars where no such motion is detected, 
and whose proximity to each other may possibly arise from casual 
'uxta-position, or from one being in the range of the other. Be- 

* During his life he observed in all, 2400 double stars. 

t Phil. Trans. 17821785. t Baily, Astron. Trans. 11. 542. 

$ The Catologue of Struve, contains 3063. 



tween fifty and sixty instances of changes to a greater or less 
amount of the relative position of double stars, are mentioned by 
Sir William Herschel ; and a few of them had changed their places 
so much within 25 years, and in such order, as to lead him to the 
conclusion that they perform revolutions, one around the other, 
in regular orbits. 

425. These conclusions have been fully confirmed by later ob- 
servers, so that it is now considered as fully established, that there 
exist among the fixed stars, binary systems, in which two stars 
perform to each other the office of sun and planet, and that the 
periods of revolution of more than one such pair have been ascer- 
tained with something approaching to exactness. Immersions and 
emersions of stars behind each other have been observed, and real 
motions among them detected rapid enough to become sensible 
and measurable in very short intervals of time. The following 
table exhibits the present state of our knowledge on this subject. 


Period in years. 

Major axis of the orbit. 


v\ Corona?, 
I Ursae Majoris, 
70 Ophiuchi, 
(f Coronae, 
61 Cygni, 
y Virginis, 
y Leonis, 





From this table it appears (1) that the periods of the double stars 
are very various, ranging in the case of those already ascertained 
from forty three years to one thousand ; (2) that their orbits are 
very small ellipses more eccentric than those of the planets, the 
greatest of which (that of Mercury) having an eccentricity of only 
about .2 of the major axis. 

The most remarkable of the binary stars is y Virginis, on account 
not only of the length of its period, but also of the great diminution 
of apparent distance, and rapid increase of angular motion about 
each other of the individuals composing it. It is a bright star 

According to E. P. Mason, 171 years. 


of the fourth magnitude, and its component stars are almost ex- 
actly equal. It has been known to consist of two stars since the 
beginning of the eighteenth century, their distance being then 
between six and seven seconds ; so that any tolerably good tele- 
scope would resolve it. Since that time, they have been con- 
stantly approaching, and were in 1838 hardly more than a single 
second asunder ; so that no telescope that is not of a very supe- 
rior quality, is competent to show them otherwise than as a single 
star, somewhat lengthened in one direction. It fortunately hap- 
pens that Bradley (Astronomer Royal) in 1718, noticed, and re- 
corded in the margin of one of his observation books, the apparent 
direction of their line of junction, as being parallel to' that of two 
remarkable stars a and 5 of the same constellation, as seen by the 
naked eye, a remark which has been of signal service in the 
investigation of their orbit. It is found that it passed its perihelion 
in June, 1836. The period given in the table is that assigned 
by Sir John Herschel ; but later observations indicate a much 
shorter period. In the interval from 1839 to 1841, the star g Ursae 
Majoris, completed a full revolution from the epoch of the first 
measurement of its position in 1781 ; and the regularity with which 
it has maintained its motion, is said to have been exceedingly 

426. The revolutions of the binary stars have assured us of 
that most interesting fact, that the law of gravitation extends to 
the fixed stars. Before these discoveries, we could not decide 
except by a feeble analogy that this law transcended the bounds 
of the solar system. Indeed, our belief of the fact rested more 
upon our idea of unity of design in all the works of the Creator, 
than upon any certain proof; but the revolution of one star around 
another in obedience to forces which must be similar to those that 
govern the solar system, establishes the grand conclusion, that the 
law of gravitation is truly the law of the material universe. 

We have the same evidence (says Sir John Herschel) of the 
revolutions of the binary stars about each other, that we have of 
those of Saturn and Uranus about the sun ; and the correspond* 

* Ast. Trans, v, 35. 


ence between their calculated and observed places in such elon- 
gated ellipses, must be admitted to carry with it a proof of the 
prevalence of the Newtonian law of gravity in their systems, of 
the very same nature and cogency as that of the calculated and 
observed places of comets round the center of our own system. 

But (he adds) it is not with the revolutions of bodies of a plan- 
etary or cometary nature round a solar center that we are now 
concerned ; it is with that of sun around sun, each, perhaps, ac- 
companied with its train of planets and their satellites, closely 
shrouded from our view by the splendor of their respective suns, 
and crowded into a space, bearing hardly a greater proportion to 
the enormous interval which separates them, than the distances of 
the satellites of our planets from their primaries, bear to their dis- 
tances from the sun itself. 

427. Some of the fixed stars appear to have a real motion in space. 

The apparent change of place in the stars arising from the pre- 
cession of the equinoxes, the nutation of the earth's axis, the dimi- 
nution of the obliquity of the ecliptic, and the aberration of light, 
have been already mentioned ; but after all these corrections are 
made, changes of place still occur, which cannot result from any 
changes in the earth, but must arise from changes in the stars 
themselves. Such motions are called the proper motions of the 
stars. Nearly 2000 years ago, Hipparchus and Ptolemy made the 
most accurate determinations in their power of the relative situa- 
tions of the stars, and their observations have been transmitted to us 
in Ptolemy's Almagest ; from which it appears that the stars retain 
at least very nearly the same places now as they did at that period. 
Still, the more accurate methods of modern astronomers, have 
brought to light minute changes in the places of certain stars 
which force upon us the conclusion, either that our solar system 
causes an apparent displacement of certain stars, by a motion of its 
own in space, or that they have themselves a proper motion. Pos- 
sibly, indeed, both these causes may operate. 

428. If the sun, and of course the earth which accompanies 
him, is actually in motion, the fact 'may become manifest from 
the apparent approach of the stars in the region which he is leav- 


ing, and the recession of those which lie in the part of the heav- 
ens towards which he is travelling. Were two groves of trees 
situated on a plain at some distance apart, and we should go from 
one to the other, the trees before us would gradually appear fur- 
ther and further asunder, while those we left behind would appear 
to approach each other. Some years since, Sir William Herschel 
supposed he had detected changes of this kind among two sets of 
stars in opposite points of the heavens, and announced that the 
solar system was in motion towards a point in the constellation 
Hercules ;* but other astronomers have not found the changes 
in question such as would correspond to this motion, or to any 
motion of the sun ; and while it is a matter of general belief that 
the sun has a motion in space, the fact is not considered as yet 
entirely proved. 

429. In most cases where a proper motion in certain stars has 
been suspected, its annual amount has been so small, that many 
years are required to assure us, that the effect is not owing to 
some other cause than a real progressive motion in the stars them- 
selves ; but in a few instances the fact is too obvious to admit of 
any doubt. Thus the two stars 61 Cygni, which are nearly equal, 
have remained constantly at the same, or nearly at the same dis- 
tance of 15" for at least fifty years past. Meanwhile they have 
shifted their local situation in the heavens, 4' 23", the annual 
proper motion of each star being 5."8, by which quantity this 
system is every year carried along in some unknown path, by a 
motion which for many centuries must be regarded as uniform 
and rectilinear. A greater proportion of the double stars than 
of any other indicate proper motions, especially the binary stars 
or those which have a revolution around each other. Among 
stars not double, and no way differing from the rest in any other 
obvious particular, ^ Cassiopeise has the greatest proper motion of 
any yet ascertained, amounting to nearly 4" annually. 


430. We cannot ascertain the actual distance of any of the fixed 
* Phil. Trans. 1783, 1805, and 1806. t See Article V. of the Addenda. 


stars, but can certainly determine that the nearest star is more than 
(20,000,000,000,000,) twenty billions of miles from the earth. 

For all measurements relating to the distances of the sun and 
planets, the radius of the earth furnishes the base line (Art. 87.) 
The length of this line being known, and the horizontal parallax 
of the body whose distance is sought, we readily obtain the dis- 
tance by the solution of a right angled triangle. But any star 
viewed from the opposite sides of the earth, would appear from 
both stations, to occupy precisely the same situation in the celes- 
tial sphere, and of course it would exhibit no horizontal parallax. 

But astronomers have endeavored to find a parallax in some of 
the fixed stars by taking the diameter of the earths orbit as a base 
line. Yet even a change of position amounting to 190 millions 
of miles, proves insufficient to alter the place of a single star, from 
which it is concluded that the stars have not even any annual par 
allax ; that is, the angle subtended by the semi-diameter of the 
earth's orbit, at the nearest fixed star, is insensible. The errors to 
which instrumental measurements are subject, arising from the 
defects of the instruments themselves, from refraction, and from 
various other sources of inaccuracy, are such, that the angular 
determinations of arcs of the heavens cannot be relied on to less 
than 1". But the change of place in any star when viewed at 
opposite extremities of the earth's orbit, is less than 1", and there- 
fore cannot be appreciated by direct measurement. It follows, 
that, when viewed from the nearest star, the diameter of the earth's 
orbit would be insensible ; the spider line of the telescope would 
more than cover it. 

431. Taking, however, the annual parallax of a fixed star at 1", 
let a b (Fig. 73) represent the radius of the earth's orbit and c a 
fixed star, the angle at c being 1 " and the angle at b a right angle ; 

Sin. 1" : Rad.::l : 200,000, nearly. 

Hence the hypothenuse of a triangle whose vertical angle is 
1" is about 200,000 times the base; consequently the distance of 
the nearest fixed star must exceed 95000000 x 200000=190000000 x 
100000, or one hundred thousand times one hundred and ninety 
millions of miles. Of a distance so vast we can form no adequate 


conceptions, and even seek to measure it only by the time 
that light, (which moves more than 192,000 miles per ' c 
second and passes from the sun to the earth in 8m. 13.3 
sec.,) would take to traverse it, which is found to be more 
than three and a half years. 

If these conclusions are drawn with respect to the 
largest of the fixed stars, which we suppose to be vastly 
nearer to us than those of the smallest magnitude, the idea 
of distance swells upon us when we attempt to estimate 
the remoteness of the latter. As it is uncertain, however, 
whether the difference in the apparent magnitudes of the 
stars is owing to a real difference or merely to their being 
at various distances from the eye, more or less uncertainty 
must attend all efforts to determine the relative distances 
of the stars; but astronomers generally believe that the 
lower orders of stars are vastly more distant from us than the 
higher. Of some stars it is said, that thousands of years would 
be required for their light to travel down to us. 

432. We have said that the stars have no annual parallax ; yet 
it may be observed that astronomers are not exactly agreed on 
this point. Dr. Brinkley, a late eminent Irish astronomer, sup- 
posed that he had detected an annual parallax in a Lyrse amounting 
to l".13 v and in a Aquilae of I". 42.* These results were contro- 
verted by Mr. Pond of the Royal Observatory of Greenwich ; and 
Mr. Struve of Dorpat has shown that in a number of cases, the 
parallax is negative, that is, in a direction opposite to that which 
would arise from the motion of the earth. Hence, until recently, 
it was considered doubtful whether in all cases of an apparent 
parallax, the effect is not wholly due to errors of observation. 

433. Indirect methods have been proposed for ascertaining the 
parallax of the fixed stars by means of observations on the double 
stars. If the two stars composing a double star are at different 
distances from us, parallax would affect them unequally, and change 
their relative position with respect to each other ; and since the 

* Phil. Trans. 1821. 


ordinary sources of error arising from the imperfection of instru- 
ments, from precession, nutation, aberration, and refraction, would 
be avoided, (as they would affect both objects alike, and there- 
fore would not disturb their relative positions,) measurements 
taken with the micrometer of changes much less than 1" may be 
relied on. Sir John Herschel proposes a method* by which 
changes may be determined which amount to only +\ of a second.f 

434. The immense distance of the fixed stars is inferred also 
from the fact that the largest telescopes do not increase their ap- 
parent magnitude. They are still points, when viewed with the 
highest magnifiers, although they sometimes present a spurious 
disk, which is owing to irradiation.} 


435. The stars are bodies . greater than our earth. If this were 
not the case they could not be visible at such an immense distance. 
Dr. Wollaston, a distinguished English philosopher, attempted to esti- 
mate the magnitudes of certain of the fixed stars from the light which 
they afford. By means of an accurate photometer (an instrument 
for measuring the relative intensities of light) he compared the 
light of Sirius with that of the* sun. He next inquired how far the 
sun must be removed from us in order to appear no brighter than 
Sirius. He found the distance to be 141,400 times its present dis- 
tance. But Sirius is more than 200,000 times as far off as the sun 
(Art. 431.) Hence he inferred that, upon the lowest computation, 
Sirius must actually give out twice as much light as the sun ; or 
that, in point of splendor, Sirius must be at least equal to two 

Phil. Trans. 1826. 

t Very recent intelligence informs us, that Professor Bessel of Konigsberg, has ob- 
tained decisive evidence of an annual parallax in 61 Cygni, amounting to 0."3483. 
This makes the distance of that star, equal to 592000 times 95 millions of miles, a 
distance which it would take light 9 years to traverse. 

t Irradiation is an enlargement of objects beyond their proper bounds, in consequence 
of the vivid impression of light on the eye. It is supposed to increase the apparent di- 
ameters of the sun and moon from three to four seconds, and to create an appearance 
of a disk in a fixed star which, when this cause is removed, is seen as a mere point. See 
Richardson, Astr. Trans, v, 1. 


suns. Indeed, he has rendered it probable that the light of Sinus 
is equal to fourteen suns. 

436. The fixed stars are suns. We have already seen that they 
are large bodies; that they are immensely further off than the 
furthest planet ; that they shine by their own light : in short, that 
their appearance is, in all respects, the same as the sun would ex- 
hibit if removed to the region of the stars. Hence we infer that 
they are bodies of the same kind with the sun. 

437. We are justified therefore by a sound analogy, in concluding 
that the stars were made for the same end as the sun, namely, as 
the centers of attraction to other planetary worlds, to which they 
severally dispense light and heat. Although the starry heavens 
present, in a clear night, a spectacle of ineffable grandeur and 
beauty, yet it must be admitted that the chief purpose of the stars 
could not have been to adorn the night, since by far the greatest 
part of them are wholly invisible to the naked eye ; nor as land- 
marks to the navigator, for only a very small proportion of them 
are adapted to this purpose ; nor, finally, to influence the earth by 
their attractions, since their distance renders such an effect entirely 
insensible. If they are suns, and if they exert no important agen- 
cies upon our world, but are bodies evidently adapted to the same 
purpose as our sun, then it is as rational to suppose that they were 
made to give light and heat, as that the eye was made for seeing 
and the ear for hearing. It is obvious to inquire next, to what 
they dispense these gifts if not to planetary worlds ; and why to 
planetary worlds, if not for the use of percipient beings ? We 
are thus led, almost inevitably, to the idea of a Plurality of 
Worlds ; and the conclusion is forced upon us, that the spot which 
the Creator has assigned to us is but a humble province of his 
boundless empire.* 

* See this argument, in its full extent, in Dick's Celestial Scenery. 



438. The arrangement of all the bodies that compose the material 
universe, and their relations to each other, constitute the System of 
the World. 

It is otherwise called the Mechanism of the Heavens ; and in- 
deed in the System of the world, we figure to ourselves a machine, 
all the parts of which have a mutual dependence, and conspire to 
one great end. " The machines that are first invented (says Adam 
Smith) to perform any particular movement, are always the most 
complex ; and succeeding artists generally discover that with fewer 
wheels and with fewer principles of motion than had originally 
been employed, the same effects may be more easily produced* 
The first systems, in the same manner, are always the most com- 
plex ; and a particular connecting chain or principle is generally 
thought necessary to unite every two seemingly disjointed appear- 
ances ; but it often happens, that one great connecting principle is 
afterwards found to be sufficient, to bind together all the discord- 
ant phenomena that occur in a whole species of things." This re- 
mark is strikingly applicable to the origin and progress of systems 
of astronomy. 

439. From the visionary notions which are generally understood 
to have been entertained on this subject by the ancients, we are 
apt to imagine that they knew less than they actually did of the 
truths of astronomy. But Pythagoras, who lived 500 years before 
the Christian era, was acquainted with many important facts in 
our science, and entertained many opinions respecting the system 
of the world which are now held to be true. Among other things 
well known to Pythagoras were the following : 

1. The principal Constellations. These had begun to be formed 
in the earliest ages of the world. Several of them bearing the 
same names as at present are mentioned in the writings of Hesiod 



and Homer ; and the " sweet influences of the Pleiades" and the 
" bands of Orion," are beautifully alluded to in the book of Job. 

2. Eclipses. Pythagoras knew both the causes of eclipses and 
how to predict them ;* not indeed in the accurate manner now 
employed, but by means of the Saros (Art. 233.) 

3. Pythagoras had divined the true system of the world, hold- 
ing that the sun and not the earth, (as was generally held by the 
ancients, even for many ages after Pythagoras,) is the center 
around which all the planets revolve, and that the stars are so many 
suns, each the center of a system like our own.f Among lesser 
things, he knew that the earth is round ; that its surface is naturally 
divided into five Zones ; and that the ecliptic is inclined to the 
equator. He also held that the earth revolves daily on its axis, and 
yearly around the sun ; that the galaxy is an assemblage of small 
stars ; and that it is the same luminary, namely, Venus, that con- 
stitutes both the morning and the evening star, whereas all the an- 
cients before him had supposed that each was a separate planet, 
and accordingly the morning star w r as called Lucifer, and the 
evening star Hesperus.J He held also that the planets were in- 
habited, and even went so far as to calculate the size of some of 
the animals in the moon. Pythagoras was so great an enthusiast 
in music, that he not only assigned to it a conspicuous place in his 
system of education, but even supposed the heavenly bodies them- 
selves to be arranged at distances corresponding to the diatonic 
scale, and imagined them to pursue their sublime march to notes 
created by their own harmonious movements, called the " music 
of the spheres ;" but he maintained that this celestial concert, 
though loud and grand, is not audible to the feeble organs of man, 
but only to the gods. 

440. With few exceptions, however, the opinions of Pythago- 
ras on the System of the World, were founded in truth. Yet they 
were rejected by Aristotle and by most succeeding astronomers 
down to the time of Copernicus, and in their place was substituted 

* Long's Astronomy, 2. 671. 

t Library of Useful Knowledge, History of Astronomy. 

t Long'a Ast. 2. 673. Ed. Encyclopaedia. 


the doctrine of Cyrstalline Spheres, first taught by Eudoxus. Ac- 
cording to this system, the heavenly bodies are set like gems in 
hollow solid orbs, composed of crystal so pellucid that no anterior 
orb obstructs in the least the view of any of the orbs that lie behind 
it. The sun and the planets have each its separate orb ; but the 
fixed stars are all set in the same grand orb ; and beyond this is 
another still, the Primum Mobile, which revolves daily from east 
to west, and carries along with it all the other orbs. Above the 
whole, spreads the Grand Empyrean, or third heavens, the abode 
of perpetual serenity.* 

To account for the planetary motions, it was supposed that each 
of the planetary orbs as well as that of the sun, has a motion of its 
own eastward, while it partakes of the common diurnal motion of 
the starry sphere. Aristotle taught that these motions are effected 
by a tutelary genius of each planet, residing in it, and directing its 
motions, as the mind of man directs his motions. 

441. On coming down to the time of Hipparchus, who flourished 
about 150 years before the Christian era,*we meet with astrono- 
mers who acquired far more accurate knowledge of the celestial 
motions. Hipparchus was in possession of instruments for meas- 
uring angles, and knew how to resolve spherical triangles. He 
ascertained the length of the year within 6m. of the truth. He 
discovered the eccentricity of the solar orbit, (although he supposed 
the sun actually to move uniformly in a circle, but the earth to be 
placed out of the center,) and the positions of the sun's apogee and 
perigee. He formed very accurate estimates of the obliquity of 
the ecliptic and of the precession of the equinoxes. He computed 
the exact period of the synodic revolution of the moon, and the 
inclination of the lunar orbit ; discovered the motion of her node 
and of her line of apsides ; and made the first attempts to ascer- 
tain the horizontal parallaxes of the sun and moon. 

Such was the state of astronomical knowledge when Ptolemy 
wrote the Almagest, in which he has transmitted to us an ency- 
clopaedia of the astronomy of the ancients. 

* Long's Ast. 2. 640 Robinson's Mcch. Phil. 2. 83 Gregory's Ast, 132 Play fair's 
Dissertation, 118. 


442. The systems of the world which have been most celebrated 
are three the Ptolemaic, the Tychonic, and the Copernican. 
We shall conclude this part of our work with a concise statement 
and discussion of each of these systems of the Mechanism of the 


443. The doctrines of the Ptolemaic System were not originated 
by Ptolemy, but being digested by him out of materials furnished 
by various hands, it has come down to us under the sanction of 
his name. 

According to this system, the earth is the center of the universe, 
and all the heavenly bodies daily revolve around it from east to 
west. In order to explain the planetary motions, Ptolemy had re- 
course to deferents and epicycles, an explanation devised by Apol- 
lonius, one of the greatest geometers of antiquity.* He conceived 
that, in the circumference of a circle, having the earth for its cen- 
ter, there moves the center of another circle, in the circumference 
of which the planet actually revolves. The circle surrounding the 
earth was called the deferent, while the smaller circle, whose center 
was always in the periphery of the deferent, was called the epi- 
cycle. The motion in each was supposed to be uniform. Lastly, 
it was conceived that the motion of the center of the epicycle in 
the circumference of the deferent, and of tlie deferent itself, are 
in opposite directions, the first being towards the east, and the 
second towards the west. 

444. But these views will be better understood from a diagram. 
Therefore, let ABC (Fig. 74,) represent the deferent, E being the 
earth a little out of the center. Let abc represent the epicycle, 
having its center at v, on the periphery of the deferent. Con- 
ceive the circumference of the deferent to be carried about the 
earth every twenty-four hours in the order of the letters ; and at 
the same time, let the center v of the epicycle abed, have a slow 
motion in the opposite direction, and let a body revolve in this cir- 

* Playfair, Dissertation Second, 119. 


cle in the direction alcd. Then it will be seen that the body 
would actually describe the looped curves klmnop ; that it would 

(Fig. 74.) 

appear stationary at Z and m, and at n and o; that its motion 
would be direct from 7c to /, and then retrograde from I to m ; di- 
rect again from m to n, and retrograde from n to o. 

445. Such a deferent and epicycle may be devised for each 
planet as will fully explain all its ordinary motions ; but it is in- 
consistent with the phases of Mercury and Venus, which being be- 
tween us and the sun on both sides of the epicycle, would present 
their dark sides towards us in both these positions, whereas at one 
of the conjunctions they are seen to shine with full face.* It is 
moreover absurd to speak of a geometrical center, which has no 
bodily existence, moving around the earth on the circumference 
of another circle ; and hence some suppose that the ancients 
merely assumed this hypothesis as affording a convenient geome- 
trical representation of the phenomena, a diagram simply, with- 
out conceiving the system to have any real existence in nature. 

* Vince's Complete System, I, 96. 


446. The objections to the Ptolemaic system, in general, are the 
following : First, it is a mere hypothesis, having no evidence in its 
favor, except that it explains the phenomena. This evidence is 
insufficient of itself, since it frequently happens that each of two 
hypotheses, directly opposite to each other, will explain all the 
known phenomena. But the- Ptolemaic system does not even do 
this, as it is inconsistent with the phases of Mercury and Venus, 
as already observed. Secondly, now that we are acquainted with 
the distances of the remoter planets, and especially of the fixed 
stars, the swiftness of motion implied in a daily revolution of the 
starry firmament around the earth, renders such a motion wholly 
incredible. Thirdly, the centrifugal force that would be generated 
in these bodies, especially in the sun, renders it impossible that 
they can continue to revolve around the earth as a center. 

These reasons are sufficient to show the absurdities of the 
Ptolemaic System of the World. . 


447. Tycho Brahe, like Ptolemy, placed the earth in the center 
of the universe, and accounted for the diurnal motions in the same 
manner as Ptolemy had done, namely, by an actual revolution of 
the whole host of heaven around the earth every twenty-four 
hours. But he rejected the scheme of deferents and epicycles, and 
held that the moon revolves about the earth as the center of her 
motions ; that the sun, and not the earth, is the center of the plan- 
etary motions ; and that the sun accompanied by the planets 
moves around the earth once a year, somewhat in the manner that 
we now conceive of Jupiter and his satellites as revolving around 
the sun. 

448. The system of Tycho serves to explain all the common 
phenomena of the planetary motions, but it is encumbered with 
the same objections as those that have been mentioned as resting 
against the Ptolemaic system, namely, that it is a mere hypothesis ; 
that it implies an incredible swiftness in the diurnal motions ; and 
that it is inconsistent with the known laws of universal gravitation. 


But if the heavens do not revolve, the earth must, and this brings 
us to the system of Copernicus. 


449. Copernicus was born at Thorn in Prussia in 1473. The 
system that bears his name was the fruit of forty years of intense 
study and meditation upon the celestial motions. As already men- 
tioned, (Art. 6,) it maintains (1) That the apparent diurnal motions 
of the heavenly bodies, from east to west, is owing to the real 
revolution of the earth on its own axis from west to east ; and (2) 
That the sun is the center around which the earth and planets all 
revolve from west to east. It rests on the following arguments : 

450. First, the earth revolves on its own axis. 

1. Because this supposition is vastly more simple. 

2. It is agreeable to analogy, since all the other planets that 
afford any means of determining the question, are seen to revolve 
on their axes. 

3. The spheroidal figure of the earth, is the figure of equilibrium, 
that results from a revolution on its axis. 

4. The diminished weight of bodies at the equator, indicates a 
centrifugal force arising from such a revolution. 

5. Bodies let fall from a high eminence, fall eastward of their 
base, indicating that when further from the center of the earth 
they were subject to a greater velocity, which in consequence of 
their inertia, they do not entirely lose in descending to the lower 

451. Secondly, the planets, including the earth, revolve about the 

1. The phases of Mercury and Venus are precisely such, as would 
result from their circulating around the sun in orbits within that 
of the earth ; but they are never seen in opposition, as they would 
be if they circulated around the earth. 

2. The superior planets do indeed revolve around the earth ; 
but they also revolve around the sun, as is evident from their phases 

* Biot. 


and from the known dimensions of their orbits ; and that the sun 
and not the earth, is the center of their motions, is inferred from 
the greater symmetry of their motions as referred to the sun than 
as referred to the earth, and especially from the laws of gravitation, 
which forbid our supposing that bodies so much larger than the 
earth, as some of these bodies are, can circulate permanently around 
the earth, the latter remaining all the while at rest. 

3. The annual motion of the earth itself is indicated also by the 
most conclusive arguments. For, first, since all the planets with 
their satellites, and the comets, revolve about the sun, analogy 
leads us to infer the same respecting the earth and its satellite. 
Secondly, the motions of the satellites, as those of Jupiter and 
Saturn, indicate that it is a law of the solar system that the smaller 
bodies revolve about the larger. Thirdly, on the supposition that 
the earth performs an annual revolution around the sun. it is em- 
braced along with the planets, in Kepler's law, that the squares of 
the times are as the cubes of the distances ; otherwise, it forms an 
exception, and the only known exception to this law. Lastly, the 
aberration of light affords a sensible proof of the motion of the 
earth, since that phenomenon indicates both a progressive motion 
of light, and a motion of the earth from west to east. (Art. 195.) 

452. It only remains to inquire, whether there subsist higher 
orders of relations between the stars themselves. 

The revolutions of the binary stars (Art. 424) afford conclusive 
evidence of at 'least subordinate systems of suns, governed by the 
the same laws as those which regulate the motions of the solar 
system. The nebulce also compose peculiar systems, in which the 
members are evidently bound together by some common relation. 

In these marks of organization, of stars associated together in 
clusters, of sun revolving around sun, and of nebulae disposed 
in regular figures, we recognize different members of some grand 
system, links in one great chain that binds together all parts of 
the universe ; as we see Jupiter and his satellites combined in one 
subordinate system, and Saturn and his satellites in another, each 
a vast kingdom, and both uniting with a number of other indi- 
vidual parts to compose an empire still more vast. 


453. This fact being now established, that the stars are immense 
bodies like the sun, and that they are subject to the laws of gravi- 
tation, we cannot conceive how they can be preserved from falling 
into final disorder and ruin, unless they move in harmonious con- 
cert like the members of the solar system. Otherwise, those that 
are situated on the confines of creation, being retained by no forces 
from without, while they are subject to the attraction of all the 
bodies within, must leave their stations, and move inward with 
accelerated velocity, and thus all the bodies in the universe would 
at length fall together in the common center of gravity. The 
immense distance at which the stars are placed from each other, 
would indeed delay such a catastrophe ; but such must be the ulti- 
mate tendency of the material world, unless sustained in one har- 
monious system by nicely adjusted motions.* To leave entirely 
out of view our confidence in the wisdom and preserving goodness 
of the Creator, and reasoning merely from what we know of the 
stability of the solar system, we should be justified in inferring, 
that other worlds are not subject to forces which operate only to 
hasten their decay, and to involve them in final ruin. 

We conclude, therefore, that the material universe is one great 
system ; that the combination of planets with their satellites con- 
stitutes the first or lowest order of worlds ; that next to these 
planets are linked to suns ; that these are bound to other suns, 
composing a still higher order in the scale of being ; and, finally, 
that all the different systems of worlds, move around their common 
center of gravity, f 

* Robison's Physical Astronomy. t See Article V. of the Addenda. 



The very liberal patronage which this work has received, enables us 
to add a few articles of interest and importance not introduced into the 
earlier editions. 



THE remarkable exhibitions of shooting stars which have occurred 
within a few years past, have excited great interest among astronomers, 
and led to some new views respecting the construction of the solar sys- 
tem. Their attention was first turned towards this subject by the great 
meteoric shower of November 13th, 1833. On that morning, from two 
o'clock until broad daylight, the sky being perfectly serene and cloudless, 
the whole heavens were lighted with a magnificent display of celestial 
fireworks. At times, the air was filled with streaks of light, occasioned 
by fiery particles darting down so swiftly as to leave the impression of 
their light on the eye, (like a match ignited and whirled before the face,) 
and drifting to the northwest like flakes of snow driven by the wind ; 
while, at short intervals, balls of fire, varying in size from minute points 
to bodies larger than Jupiter and Venus, and in a few instances as large 
as the full moon, descended more slowly along the arch of the sky, often 
leaving after them long trains of light, which were, in some instances, 
variegated with different prismatic colors. 

On tracing back the lines of direction in which the meteors moved, it 
was found that they all appeared to radiate from the same point, which 
was situated near one of the stars (Gamma Leonis) of the sickle, in the 
constellation Leo ; and, in every repetition of the meteoric shower of No- 
vember, the radiant point has occupied nearly the same situation. 

This shower pervaded nearly the whole of North America, having ap- 
peared in almost equal splendor from the British possessions on the north, 
to the West India islands and Mexico on the south, and from sixty-one de- 
grees of longitude east of the American coast, quite to the Pacific ocean 
on the west. Throughout this immense region, the duration was nearly 
the same. The meteors began to attract attention by their unusual fre- 
quency and brilliancy, from nine to twelve o'clock in the evening ; were 
most striking in their appearance from two to four ; arrived at their maxi- 
mum, in many places, about four o'clock ; and continued until rendered 
invisible by the light of day. The meteors moved in right lines, or in 


such apparent curves, as, upon optical principles, can be resolved into 
right lines. Their general tendency was towards the northwest, although 
by the effect of perspective they appeared to move in all directions. 

Soon after this occurrence, it was ascertained that a similar meteoric 
shower had appeared in 1799, and what was remarkable, almost exactly 
at the same time of year, namely, on the morning of the 12th of Novem- 
ber; and it soon appeared, by accounts received from different parts of 
the world, that this phenomenon had occurred on the same 13th of No- 
vember, in 1830, 1831, and 1832. Hence, this was evidently an event 
independent of the casual changes of the atmosphere ; for, having a pe- 
riodical return, it was undoubtedly to be referred to astronomical causes, 
and its recurrence, at a certain definite period of the year, plainly indi- 
cated some relation to the revolution of the earth around the sun. 

It remained, however, to develop the nature of this relation, by inves- 
tigating, if possible, the origin of the meteors. The views to which the 
author of this work was led, suggested the probability that the same phe- 
nomenon would recur at the corresponding seasons of the year for at least 
several years afterwards ; and such proved to be the fact, although the 
appearances, at every succeeding return, were less and less striking, until 
1839, when, so far as is known, they ceased altogether. 

Meanwhile, three other distinct periods of meteoric showers have also 
been determined ; one about the 21st of April, another on the 9th of Au- 
gust, and another on the 7th of December. 

The following conclusions respecting the meteoric shower of November, 
are believed to be well established, and most of them are now generally 
admitted by astronomers, though we cannot here exhibit the evidence on 
which they were founded, but must beg leave to refer the reader to vari- 
ous publications on this subject in the American Journal of Science, com- 
mencing with the 25th volume ; and also to " Letters on Astronomy," by 
the author of this work. 

It is considered, then, as established, that the meteors had their origin 
beyond the limits of the atmosphere, having descended to us from some 
body existing in space independent of the earth ; that they consisted of 
exceedingly light combustible matter ; that they moved with very great 
velocities, amounting in some instances to not less than fourteen miles 
per second ; that some of them were bodies of large size, probably seve- 
ral hundred fe'et in diameter ; that when they entered the atmosphere, 
they rapidly and powerfully condensed the air before them, and thus 
elicited the heat which set them on fire, as a spark is sometimes evolved 
by condensing air suddenly by a piston and cylinder ; and that they were 
consumed and resolved into small clouds at the height of about thirty 
miles above the earth. 


Calling the body from which the meteors descended the " meteoric 
body," it is inferred that it is a body of great extent, since, without appa- 
rent exhaustion, it has been able to afford such copious showers of mete- 
ors at so many different times ; and hence we regard the part that has 
descended to the earth only as the extreme portions of a body or collection 
of meteors, of unknown extent, existing in the planetary spaces. Since 
the earth fell in with the meteoric body, in the same part of its orbit for 
several years in succession, the body must either have remained there 
while the earth was performing its whole revolution around the sun, or it 
must itself have had a revolution, as well as the earth. No body can 
remain stationary within the planetary spaces ; for, unless attracted to 
some nearer body, it would be drawn directly towards the sun, and 
could not have been encountered by the earth again in the same part of 
her orbit. Nor can any mode be conceived in which this event could 
have happened so many times in regular succession, unless the body had 
a revolution of its own around the sun. Finally, to have come into con- 
tact with the earth at the same part of her orbit, in two or more successive 
years, the body must have either a period which is nearly the same with 
the earth's period, or some aliquot part of it. No period will fulfil the 
conditions, but either a year or half a year. Which of these is the true 
period of the meteoric body, is not fully determined. 



(From Pont^coulant's Trait6 Etementaire de Physique Celeste, p. 503.) 

" Mathematical analysis has furnished to geometers all the resources 
which were necessary in order to rise from the consideration of the laws 
of motion of a single material point, to that of the motions of a system of 
bodies connected together in any manner whatsoever. It would be dif- 
ficult, without the help of the calculus, to give even an idea of the means 
which have been employed for their discovery ; but there have been de- 
duced from analytical formulas several general principles, which mani- 
fest themselves in the motions of every system of bodies, and which it is 
important to make known, since they have an immediate application to 
the constitution of our planetary system. These general principles of 
motion are four in number, and may be enumerated as follows : 

"1. If a system of bodies act on one another in any manner whatso. 


ever, and are subject to the action of a force directed towards a fixed cen- 
ter, and are not acted on by any extraneous force, the sum of the areas 
traced by the radius- vectors of the different bodies of the system on an 
immovable plane passing through the fixed point, multiplied respectively 
by the masses of these bodies, will be proportional to the time. This 
theorem constitutes the principle of the CONSERVATION OF AREAS. 

" If all the bodies are subject to no other force than their mutual action, 
and there were no center of action, we could then choose arbitrarily the 
origin of the radius-vectors, and the preceding principles would be true 
for all points of space. 

11 2. If a system of bodies are subject to no other force than the mutu- 
al action of all the parts, and are solicited by no extraneous force, the 
common center of gravity moves in a right line, and uniformly ; that is 
to say, its motion will be the same as if it obeyed no other force than the 
primitive impulse communicated to it ; so that, in the same manner as by 
the law of inertia, where a material point cannot, without the intervention 
of some extraneous force, change the motion it has received, likewise a 
system of bodies cannot alter the motion of its center of gravity by the 
single action of its members upon one another. This result constitutes 


" In general, whatever may be the forces which act on a system of bo- 
dies, the motion of the center of gravity of the system is the same as 
though all the bodies were united in it, and all the forces which solicit the 
bodies were applied to it directly. 

" 3. The mass of a body multiplied by the square of its velocity is 
called its active force. If the system under consideration is subject only 
to the mutual action of all its members, and to attractions directed towards 
fixed centers, the sum of the active forces of the bodies which compose 
the system, or the entire active force of the system, is constant in the case 
even where several of the bodies are compelled to move in the direction of 
given lines, or on given surfaces. This principle is that of the CONSER- 

" 4. The sum of the active forces of any system of bodies, during the 
time it employs in passing from one position to another, is a minimum. 
This law, which was for a long time attempted to be derived from meta- 
physical considerations, by supposing that nature always employs in the 
effects which she produces the least possible efforts, results directly from 
analytic formulce, and is called the PRINCIPLE OF LEAST ACTION. 

" The principle of the conservation of areas holds good whatever may 
be the direction of the plane which is chosen for the plane of projection ; 
but among all the planes which can be drawn through a given point, there 


is always one relatively to which the sum of the areas traced by the projec- 
tions of the radius-vectors, multiplied respectively by the masses of the 
bodies, is a maximum. This plane possesses a very remarkable proper- 
ty, namely, that this sum is nothing in respect to every plane which is 
perpendicular to it. This plane, moreover, always remains parallel to 
itself, whatever may be the particular motions of each of the bodies of 
the system. Hence its position can always be found, a consideration 
which renders the principle one of great value in the theory of the sys- 
tem of the world. 

" These four principles hold good in the displacements of every system of 
bodies, but it is especially in the motions of the heavenly bodies that their 
truth is manifested by an admirable agreement of the results of calcula- 
tion with those of observation. The reason is, that the disturbing causes, 
such as friction, resistance of the air, &c., which affect the motions of 
bodies observed on the earth, and which render their mathematical calcu- 
lation almost impossible, disappear, or become nearly insensible, in re- 
spect to the heavenly bodies circulating in the immensity of space, where 
we observe nothing but the effects of the principal forces which animate 

An able writer* offers the following striking illustration of the Princi- 
ple of Least Action 

" Throughout inanimate nature, all is done with the least possible ac- 
tion ; no development of force, however minute, is thrown away. If I 
wished to ascend or descend a hill, or pass from one portion of it to anoth- 
er, with the least possible muscular force, a slight consideration would 
show me that the precise path to be pursued, would be dependent on the 
form and inclination of the different parts of the hill ; upon the nature of 
my own muscular energies ; and upon other data, of which I could 
scarcely by any possibility acquire a knowledge, and on which, when 
known, my intellectual powers would be quite insufficient to enable me 
to found a conclusion. Under these circumstances, the chances are in- 
finitely greater that I should select the wrong than the right path. Now, 
if I am to project a stone up the hill, or obliquely across it, or suffer it to 
roll down it, whatever obstacles opposed its motion, whether they arose 
from friction, resistance, or any other cause, constant or casual, still 
would the stone, when left to itself, ever pursue that path in which there 
was the least possible expenditure of its efforts; and if its path were fix- 
ed, then would its efforts be the least possible in that path. This extra- 
ordinary principle is called that of least action ; its existence and univer- 
sal prevalence admit of complete mathematical demonstration. Every 
particle of dust blown about in the air, every particle of that air itself, 

* Mosely, " Mechanics applied to the Arts," Introduction, p. xxx. 


has its motions subjected to this principle. Every ray oflight that passes 
from one medium into another, deflects from its rectilinear course, that it 
may choose for itself the path of least possible action ; and for a similar 
reason, in passing through the atmosphere, it bends itself in a particular 
curve down to the eye. The mighty planets, too, that make their cir- 
cuits ever within those realms of space which we call our system ; the 
comets, whose path is beyond it ; all these are alike made to move so as 
best to economize the forces developed in their progress." 


(See Engraving, Plate I.) 

On the 28th of February, 1843, the attention of numerous observers, 
in various parts of the world, was arrested by a comet, seen in the broad 
light of day, a little eastward of the sun. On the day previous, indeed, 
it was first noticed by Captain Ray, at Conception in South America 
time, 11 o'clock, A.M. Of the observations made upon this remarka- 
ble body on the 28th, the most accurate are believed to be those made at 
Portland, Maine, by Mr. F. G. Clarke. The time of observation was 
3h. 2m. 15s., mean solar time, and the observed distance, which Mr. 
Clarke thinks may be depended on to 10", was 4 6' 15", from the farthest 
limb of the sun to the nearest limb of the comet. It resembled a white 
cloud of great density, being nearly equally brilliant throughout its whole 
length, which at this time was estimated by different observers at about 
3. During the first week in March, the appearance of the comet in 
the southern hemisphere was splendid and magnificent, enhanced, in both 
respects, by the transparency of a tropical sky, and the higher angle of 
elevation above that at which it was seen by northern observers. At 
Pernambuco, S. A., on the 4th of March, it presented a golden hue ; and 
it was described by the commander of a ship as so brilliant as to throw a 
strong light on the sea. 

At New Haven, it was first seen after sunset on the 5th of March, and 
by the writer on the 6th. It then lay far in the southwest. On account 
of the presence of the moon, it was not seen most favorably until the 
evening of the 17th. It then extended along the constellation Eridanus 
to the ears of the Hare, towards Sirius, about 40 in length, slightly curv- 
ed like a goose-quill, and colored with a slight tinge of rose red, which 
in a few evenings disappeared, and the comet afterwards appeared nearly 


white. Our diagram presents a pretty accurate idea of its appearance 
on the 20th of March. Its nucleus was at this time near the star Zibal, 
in Eridanus, (R. A. 46 4' 38".4, Dec. S. 9 9' 45".5) and it extended 
nearly parallel to the equator along the constellation Eridanus, through 
the ears of the Hare, beneath the feet of Orion, terminating near Sirius. 
The several stars represented in or near the comet, in the annexed dia- 
gram, may be easily identified by the aid of a celestial globe, or a map 
of the stars. 

All the astronomers of the age have agreed in the opinion, that this is 
one of the most remarkable exhibitions of a comet ever witnessed, al- 
though they are not fully agreed respecting the elements of its orbit, or 
its p riodic time. Professor Pierce maintains that the most probable 
period is 175 years, and consequently that the present is its first return 
since 1668; but Messrs. Walker and Kendall are of opinion that its true 
period is 2 If years, and consequently that this is the eighth return since 
1668,* and that it will visit our sphere again in 1865. 

This comet passed its perihelion on the 27th of February, at which 
time it almost grazed the surface of the sun, approaching nearer to that 
luminary than any comet hitherto observed. Its motions at this time were 
astonishingly swift, and its brilliancy such as to -induce the belie/ that it 
was at a white heat throughout its whole extent. 

According to the determination of Messrs. Lauguier and Mauvais, of 
the Paris Observatory, which is believed to be as accurate as any, the 
following are the elements of this comet 

Time of Perihelion. Lon. Per. Lon. As. Node. Inclination. Per. Diet. Course. 

27d.42'291 278 45' 39" 2 10' 0" 35 31' 30" 0.005488 Ret. 


(See Engraving, Plate II.) 

As an accurate representation of these delicate and interesting objects 
requires the most finished engraving and printing, and cannot be well 
executed in the body of the work, we annex a separate cut giving 
a correct view of several of these remarkable objects. The double 
stars were figured by the late Ebenezer Porter Mason, and the nebu- 
la is from the Glasgow edition of Nichols's " Architecture of the 
Heavens" a work which contains a great variety of elegant and accu- 
rate delineations of stars and nebulae. 

* See American Almanac, 1844, p. 94. American Journal of Science, xlv. 188- 





If we contemplate the relations subsisting between a central body, as 
the sun, and a revolving body, as one of the planets, it will be readily 
understood, that if the quantity of matter in the central body is in- 
creased, while the distance of the revolving body remains the same, 
the velocity of the revolving body must be increased also, in order to 
generate a sufficient centrifugal force to counterbalance the increased 
force of attraction in the central body, arising from the increase of its 
mass ; and that, were the force of attraction diminished by removing 
the body to a greater distance from the center, then the rate of its mo- 
tion would also have to be diminished, otherwise the centrifugal force 
would overpower the force of attraction. It is a remarkable fact, that 
the members of the solar system are so adjusted to each other, in re- 
spect to their velocities, distances from the sun, periodic times, and 
gravitation towards the central body, that if any one of these particulars 
is known, all the rest become known also. Thus, if it were found that 
a new-discovered planet moved in its orbit six times as slow as the 
earth, we should know at once that its distance from the sun was thirty- 
six times as great as the earth's distance, that its time of revolution was 
two hundred and sixteen years, and that its gravitation towards the sun 
was twelve hundred and ninety-six times less than that of the earth ; 
for the distance is the square of the number expressing the rate of mo- 
tion compared with that of the body taken as a standard ; the periodic 
time is the cube ; and the gravitation to the sun is the biquadrate of the 
same number. All this follows from Kepler's third law that the 
squares of the periodic times are as the cubes of the distances; and 
from the law of universal gravitation that the force of attraction is in- 
versely as the square of the distance. The four particulars named, 
therefore, constitute a series of numbers in geometrical progression, of 
which the first term is equal to the ratio. The truth of this proposition 
may be demonstrated as follows. 

*< In the preparation of this article, the author has derived much assistance from a 
small work, now nearly out of print, containing the substance of three lectures deliv- 
ered to the students of Yale College in 1781, by Rev. Nehemiah Strong, at that time 
Professor of Mathematics and Natural Philosophy. 



Let D be the mean distance of a planet from the sun, # the ratio of 
the diameter to the circumference of a circle, and P the time of revo- 
lution around the sun, or periodic time ; then the expression for the 

2*D D D 8 

velocity is V= x - . And V* x p- But, by Kepler's law, P 2 x D' 

D 2 1 

. . V 2 oc yp or V a x =r. Since a body more remote from the sun moves 

more slowly in its orbit than a nearer body, and the comparative slow- 
ness, or retardation, is inversely as the velocity, in order to avoid frac- 
tional terms, we may put the retardation (R) in the place of V, and then 
R 2 xD, (1.) If, therefore, R indicates how much slower a planet 
moves than another, as the earth, taken as a standard, the square of R 
will show how much farther from the sun the planet is than the earth. 

D D 3 

Again, since V oc , V 3 oc ^. But, by Kepler's law, D 3 oc P 2 .-. V 3 x 

^orV'oDp, andR 3 ocP(2.) 

Consequently, if R expresses the retardation of a planet in compari- 
son with the earth, the cube of R will express the corresponding peri- 
odic time. 

Finally, by the law of gravitation, the force of gravitation towards 
the central body varies as the square of the distance inversely, or 

G oc =-jj. But the diminution of gravity (L) being inversely as the grav- 
ity,. L x D 2 ; but D x R 9 .-. D 2 x R 4 , and L x R 4 (3.) 

Therefore, if R denotes how much slower a planet moves in its orbit 
than the earth, R 4 will denote how much less the same body gravitates 
towards the central body. Collecting these several results, it appears 
that the square of the rate of motion gives the distance, its cube the peri- 
odic time, and its fourth power the diminution of gravity, which numbers 
compose a series in geometrical progression of which the first term is the 

A number of very useful and convenient rules, may be derived from 
this numerical relation between the members of the solar system ; since, 
when any one of the four things named is given, all the rest may 
be found from it ; and each of the four may be found in four differ- 
ent ways when the other members of the series are given. This will be 
obvious from a few examples. 


1. Square the retardation for the distance. 

2. Cube the retardation for the periodic time. 

3. Take the fourth power of the retardation for the force of gravitation, 


II. Given the DISTANCE, (D.) 

1. Take the square root of the distance for the rate of motion. 

2. Take the cube of the square root of the distance for the periodic 

3. Take the square of the distance for the force of gravitation. 

III. Given the PERIODIC TIME, (P.) 

1. Take the cube root of the periodic time for the rate of motion. 

2. Take the square of the cube root of the periodic time for the dis- 

3. Take the biquadrate of the cube root for the force of gravitation. 

IV. Given the diminished FORCE OF GRAVITATION. (L.) 

1. Take the fourth root for the rate of motion. 

2. Take the square root for the distance. 

3. Take the cube of the fourth root for the periodic time. 

V. Required the RATE OF MOTION. 

This may be obtained by taking the square root of the distance, or 
the cube root of the periodic time, or the biquadrate root of the force of 
gravitation, or by dividing the force of gravitation by the periodic time. 

VI. Required the DISTANCE. 

Take the square of the retardation, or the square of the cube root of 
the time, or the square root of the force of gravitation, or divide the 
time by the retardation. 

VII. Required the PERIODIC TIME. 

We may take the cube of the retardation, or the cube of the square 
root of the distance, or the cube of the fourth root of the gravitation, or 
may divide the gravitation by the retardation. 

VIII. Required the diminished GRAVITATION. 

It may be found from the fourth power of the retardation, or the 
square of the distance, or the biquadrate of the cube root of the time, 
or by multiplying the periodic time by the retardation. 

According to the foregoing rules tables may be formed, exhibiting, in 
a striking, light, the numerical relations of the members of the solar 
system. In the following table the distances are taken from Herschel's 
Astronomy, and from these the other particulars are determined by the 
preceding rules. If Mercury were taken as the standard of comparison, 
then the retardations of all the other planets would be greater than 
unity ; but, as it is convenient to take the earth as the standard, the 
retardations of Mercury and Venus will be less than unity : showing 
that the velocity (which is expressed by the fraction inverted) is greater 
than that of the earth. In like manner, the force of gravitation of an 
inferior planet, being greater than that of the qprth, is the reciprocal 
of the tabular number. 







Per. Times. 

Force of Gravitation. 










































PROB. 1. The planet Pallas was discovered to have a period of about 
4j years. How much slower does it move in its orbit than the earth 
how much further is it from the sun and how much less does it grav- 
itate towards the sun ? Ans. R=1.67, D=2.79, L 7.80. 

By applying to the earth's motion per second, its distance from the 
sun in miles, and the space through which the earth departs in a 
second from a tangent to her orbit, the proportional numbers deter- 
mined by this problem, we may obtain the numerical value of each of 
these elements. 

PROB. 2. -What would be the periodical time of a meteor or planet 
revolving close to the earth ? 

As the moon is a body revolving around the earth at a known dis- 
tance, and with a known periodic time, it will evidently furnish the 
necessary standard of comparison. The distance of the moon from the 
center of the earth being 60 times the earth's radius, and of course 60 
times that of the meteor, its rate of motion is ^/ 60 times less. The 


retardation being -y/60, the periodic time will be 60 2 . Now what 
part of the moon's period is 60^ ? Divide the moon's period (27.32 

days) by 60^, and we have for the answer 1 hour, 24 minutes, 38.88 

PROB. 3. What would be the periodic time of a body revolving 
about the earth at the distance of 5000 miles from the center ? Ans. 
Ih. 59m. 23.28s. 

PROB. 4. How much faster must the earth revolve in order that 
bodies on its surface may lose all their gravity ? 

According to probler$ 1, the period of a body revolving at the sur- 
face of the earth, is 1.4108 hours ; and since, in a circular orbit, the 


force of gravity and the centrifugal force are equal, therefore, a body 
like that contemplated in problem 1, is in equilibrium between these two 
forces ; consequently such a body may be considered as having lost all 
its gravity, and being, by the supposition, close to the earth, we have 
only to inquire how much its velocity exceeds that of the earth. Now 
24 divided by 1.4108 gives 17.01 ; which shows that were the earth to 
revolve on its axis about 17 times faster than it does at present, the 
bodies on the surface would lose all their weight ; and were the velocity 
greater than this, the centrifugal force would prevail over the centripe- 
tal, and the same would fly off from the earth in tangents. 

PROS. 5. Were the moon to be removed so far from the earth as to 
revolve about it but once a year, how much greater would be its dis- 
tance than at present, how much less its velocity, and its gravitation 
towards the earth ? 

Its period being increased 13.37 times, its retardation is 13.37^2.37 ; 
its distance 2.37 2 =5.633 ; and its diminished gravity 5.633 2 =31.73. 
Or R=2.37, D=5.633, and L=31.73. 

Multiplying the present distance of the moon, 238,545 miles, by 5.633, 
we obtain about 1,344,000 miles for the distance at which the moon must 
have been placed in order to complete its revolution in one year. 

PROS. 6. Were the earth's mass equal to the sun's, and of course 
354,000 times as great as at present, in what time would the moon re- 
volve around it ? 

Since, at a given distance, any increase of mass in the central body 
is accompanied by an equal increase of attraction, which must be 
balanced by a corresponding increase of velocity in the revolving body, 
so as to generate an equal centrifugal force ; and since the centrifugal 
force in a given circular orbit varies as the square of the velocity, there- 
fore V s oc 354,000, and V oc y' 354,000, and the periodic time being in- 

versely as the velocity, P -y~00 =594^= thatis > the P 6 *" 
time is diminished about 595 times ; and 27.32 divided by 595 gives Ih. 
5m. 57s. as the time in which the moon would be hurled around the 
earth, were the quantity of matter in the earth equal to that in the 

Comets, in passing their perihelion, especially when that happens to 
be very near the sun, as in the great comet of 1843, move with an as- 
tonishing rapidity ; requiring a velocity not merely sufficient to gener 
ate the centrifugal force necessary to balance the powerful force of at- 
traction exerted by the sun, but greatly to exceed that force, since they 
are carried far without a circular orbit into an elliptical or even a hy- 
periodic orbit. 


PROB. 7. How much must the mass of the earth be increased in 
order that the moon may revolve about it in the same time as at present, 
when removed to three times her present distance ? 

The quantity of matter in the central body is proportioned to the 
cube of the distance of the satellite divided by the square of the periodic 
time, (Art. 379.) Here the time being given M <r D 3 3 3 =27, which 
shows that, had the moon been placed three times as far from the earth 
as at present, she would have required 27 times as much matter to 
make the moon revolve in her present period. 

PROB. 8. How much must the mass of the earth be increased to 
make the moon, at her present distance, revolve in 24 hours ? Ans. 
746.4 times. 

PROB. 9. The semi-diameter of Jupiter being 11 times that of the 
earth, and the distance of its fourth satellite from the center of the 
planet being 27 times the radius of the planet ; also the sidereal revolu- 
tion of the satellite being 16.69 days, while that of the moon is 27.3217 
days, and her distance 60 times the radius of the earth : How much 
does the quantity of matter in Jupiter exceed that of the earth ? Ans. 
324.49 times. 

PROB. 10. Suppose volcanic matter to be thrown from the moon to- 
wards the earth, required the point where it would be in equilibrium 
between the two, the mass of the moon being one-eightieth that of the 
earth ? Ans. 24,000 miles from the center of the earth, nearly. 

PROB. 11. Suppose that the only two bodies in the universe were a 
sphere two inches in diameter, of the same density with the earth, for 
the primary, and a material point for the satellite : What would be the 
periodic time of the satellite, at the distance of one foot, in a circular 
orbit ? Ans. 2 days, 9 hours, 41 minutes. 


Within a short period astronomical science has been enriched with 
several capital discoveries, showing that neither the field of observation, 
nor the resources of mathematical analysis are exhausted, and inspiring 
the hope and belief that truths in astronomy more recondite, and no less 
wonderful than those revealed successively to preceding ages, will con. 
tinue to crown the labors of astronomers to the end of time. 


DISTANCES OF THE STARS. After many fruitless and delusory efforts 
to measure the immense interval that separates us from the fixed stars, 
the great Prussian astronomer, Bessel, in the year 1838, determined 
this interesting and important element, by observations on a double star 
in the Swan, (61 Cygni.) This star was selected for the following 
reasons : first, it was known to have, among all the stars, the greatest 
proper motion, indicating a comparatively great proximity to our system ; 
secondly, being a double star, it was peculiarly well adapted to instru- 
mental observation ; thirdly, situated as it is among the circumpolar 
stars, observations could be made upon it nearly every night in the 
year; and, finally, the great number of small stars in the imme- 
diate neighborhood, furnished the opportunity of selecting favorable 
stationary points from which (inasmuch as these more remote objects 
might be considered as entirely devoid of parallax) any changes of 
place in the nearer, in consequence of an annual parallax, might be 
readily estimated. By observations of the last degree of refinement, 
conducted for a period of several years, a parallax was decisively indi- 
cated, amounting to about one-third of a second ; or, more exactly, to 
0/'3483, implying a distance of 592,200 times the mean distance of the 
earth from the sun, or a space which it would take light, moving at the 
rate of twelve millions of miles per minute, nine and a quarter years to 
traverse.* To form some familiar notions of this distance, let us sup- 
pose a railway-car to travel night and day, at the rate of twenty miles 
an hour, we should find it would take it about 547 years to reach the 
sun ; but to reach 61 Cygni would require 324,000,000 of years. 

The observations of Bessel enabled him to estimate also the period of 
revolution of the two stars composing the binary system of 61 Cygni, 
and the dimensions of the orbit, and he found the periodic time about 
540 years, and the length of the orbit about two and a half times that 
of Uranus. Knowing also the distance of this star, we can now deter- 
mine from its proper motion (five seconds a year) the velocity of its mo- 
tion : this is found to be about forty-four miles per second, more than 
double that of the earth in its orbit amounting to about one thousand 
millions of miles per annum. 

On account of the smallness of the supposed parallax thus found, it 
would not be unreasonable still to entertain a lingering suspicion, that 
it is nothing more than the unavoidable imperfection of instrumental 
measurements, as proved to be the case in previous attempts to find 
the same element ; but the most satisfactory evidence which the world 

It will be remarked that this is the result of the second series of observations, and 
M deemed more accurate than those mentioned in the text 


can have that such is not the fact in the present instance, but that the 
parallax is truly found, is that the most celebrated astronomers of the 
age, after rigorous scrutiny, have acknowledged the reality and sound- 
ness of the determination. 

Several other stars have of late been supposed to indicate a parallax ; 
and one of them, Alpha Centauri, is thought by some to be nearer to us 
than 61 Cygni, having a parallax of nine-tenths of a second. The 
great Russian astronomer, M. Struve, has also announced a parallax in 
Alpha Lyra of a quarter of a second ; but this result has not yet been 

NEW PLANETS. Unexpectedly, four new planets have in rapid suc- 
cession been revealed to us, namely Astrsea, Iris, Hebe, and Neptune. 
The first three belong to the group of Asteroids, making the whole num- 
ber at present known, seven instead of four. They are very small 
bodies, seen by the telescope as faint stars, but are known to be planets 
from their having a motion of revolution around the sun.' 

The discovery of the planet Neptune, by a distinguished French as- 
tronomer, Le Verrier, is one of the most remarkable events in astrono- 
my ; both its existence and its position in the heavens having been re- 
vealed to a profound mathematical analysis before it was seen with the 
telescope. The method of investigation, although laborious and intri- 
cate, is not difficult to be understood, but may be described in very sim- 
ple terms. The planet Uranus has been long known to be subject to 
certain irregularities in its revolution around the sun, not accounted for 
by all the known causes of perturbation. In some cases the deviation 
from the trqe place, as given by the tables, differs from actual observa- 
tion two minutes of a degree, a quantity, indeed, which seems small, 
but which is still far greater than occurs in the case of the other planets, 
Jupiter and Saturn, for example, and far too great to satisfy the extreme 
accuracy required by modern astronomy. This long since suggested 
to astronomers the possibility of one or more additional planets, hitherto 
undiscovered, which, by their attractions, exerted on Uranus a great 
disturbing influence. 

Le Verrier, assuming the existence of such a planet, applied him- 
self, by the aid of the calculus, guided by the law of universal gravi- 
tation, to the inquiry, where is it situated at what distance from the 
sun and in what point of the starry heavens ? According to Bode's 
law of the planetary distances, (according to which Saturn is nearly 
twice as far from the sun as Jupiter, and Uranus twice as far as Saturn,) 
he inferred that, if a planet exists beyond Uranus its distance is proba- 
bly twice that of Uranus, or about thirty-six millions of miles from the 
sun, which is about thirty-eight times that of the earth. He finally 


fixed it thirty-six times the mean distance of the earth. Hence its 
periodic time would be 216 years. After reasoning from analogy, and 
the doctrine of universal gravitation, respecting the position and mass 
which a body must have in order to occasion the perturbations of Uranus 
to be accounted for, equations were formed between these perturbations 
and the elements of the body in question, both known and unknown. 
These equations involved nine unknown quantities, and their resolution 
involved difficulties almost insurmountable ; but, by the most ingenious 
artifices, the several unknown quantities were successfully eliminated, 
either directly or by repeated approximations, until he arrived at ex- 
pressions for the elements of the unknown planet, which gave its exact 
place among the stars, its quantity of matter, the shape of its orbit, and 
the period of its revolution. At the sitting of the French Academy, 
August 31, 1846, Le Verrier presented the following elements : 

Longitude of the planet, Jan. 1, 1847, - 326 32" 

Mass, that of the sun being 1, - 

Eccentricity, - 0.107 

Time of revolution, - - - 217.387 years 

Longitude of the perihelion, - - 284 45' 

Major axis of the orbit, that of the earth being 1, 36.154 

He was therefore enabled to say, that the planet was then just passing 
its opposition, and consequently was most favorably situated for obser- 
vation, and on account of the slowness of its motion, would remain in a 
very favorable position for three months afterwards. Le Verrier wrote 
to M. Galli, of Berlin, communicating his latest results, and requesting 
him to reconnoiter for the stranger, directing his telescope to a point 
about five degrees eastward of the well-known star, Delta Capricorni. 
That astronomer no sooner pointed his telescope to the region assigned, 
than he at once recognised the body, its place being only 52 minutes of 
a degree distant from the position marked out for it by Le Verrier, and 
its apparent diameter being almost the same that he had assigned. 

By a singular coincidence, a young mathematician of the University 
of Cambridge, (Eng.,) Mr. Adams, had, without the least knowledge of 
what M. Le Verrier was doing, arrived at the same great result. But 
having failed to publish his paper until the world was made acquainted 
with the facts through the other medium, he has lost much of the honor 
which the priority of discovery would have gained for him. 

Thus two distinguished mathematicians, unknown to each other, and 
by entirely independent processes, had arrived at the same results, as 
regarded both the existence of the supposed planet, and the region of 



the starry heavens where at that moment it lay concealed ; and, to 
crown all, astronomers, in obedience to the directions of one of them, 
had pointed their telescopes to the spot and found it there. The con- 
viction on the mind of every one was, that nothing but absolute truth 
could abide a test so unequivocal. It still remained, however, to deter- 
mine by observation whether the body actually conformed, in all re- 
spects, to the results of theory. To settle this point completely, that is, 
to determine with precision the elements of the orbit from observation, 
would require a long time in a planetary body whose motion was so 
slow that more than two centuries, as was supposed, would be required 
to complete a single revolution. But if it should be found, that among 
preceding catalogues of the stars this body might have been included, 
and its place recorded as a fixed star, then, by comparing that place 
with its present position, and noting the interval of time between the 
two observations, we might thus learn the rate of its motion and its 
periodic time, and might thence deduce various other particulars de- 
pendent on these elements. Our distinguished countryman, Mr. S. C. 
Walker, then connected with the observatory at Washington, undertook 
this investigation. First, from the observations already accumulated 
he calculated the path which the planet must have pursued for the last 
fifty or sixty years, and by tracirig this path among the stars of La- 
lande's catalogue, he found that it passed within two minutes of a star 
of the seventh magnitude, which was recorded as being seen in May, 
1795. Professor Hubbard, of the same observatory, on reconnoitering 
for this star, ascertained that it was missing. Little doubt remained, 
that the star seen by Lalande was the planet of Le Verrier ; and this 
conclusion was confirmed by calculating its orbit on this supposition, 
and comparing the results with the places it has actually occupied since 
it fell within the sphere of observation. 

The results thus obtained were, however, materially different from 
those of Le Verrier and Adams. Instead of a period of 216 years, as 
given by Le Verrier, they gave a period of only 166 years ; and in- 
stead of a distance of 3600 millions of miles, the new period would re- 
quire a distance of only 2364 millions. The eccentricity of the orbit, 
moreover, according to Walker, was much less than had been assigned 
to it, the orbit being, in fact, very nearly circular, while by Le Verrier's 
estimate it was considerably elliptical. The longitude, in fact, proved 
to be nearly the same as that assigned ; and hence the fortunate dis- 
covery of the body by the telescope. 

Professor Pierce, the able professor of astronomy in Harvard Univer- 
sity, has been led to the conclusion, both from his own theoretical in- 
vestigations and from the investigations of Mr. Walker, that this planet 


does not fully account for the residuary perturbations of Uranus ; and, 
therefore, that its discovery at the place assigned by Le Verrier, was 
rather the result of accident than the legitimate consequence, of his 

CENTER OF THE UNIVERSE. At the end of the preceding treatise we 
have suggested reasons for believing that the whole host of heaven re- 
volve around a common center. Dr. Mcedler, of the Imperial Observa- 
tory at Dorpat, has recently not only asserted this doctrine, but has en- 
deavored to show the exact position of that center. He fixes it in the 
Pleiades, and asserts that Alcyone, the brightest star of this group, is the 
true " central sun," around which all the stars of our visible firmament 
revolve, in obedience to the law of universal gravitation. The proofs 
of this remarkable hypothesis are deemed too incomplete, at present, to 
command entire assent ; but the method of investigation pursued by this 
distinguished astronomer, opens a new field of observation and of specu- 
lation, and promises to lend a new interest to inquiries into the mechan- 
ism of the universe. 

REVELATIONS OF THE TELESCOPE. Practical astronomy has of late 
been enriched with a number of great telescopes, which have disclosed 
new wonders in the starry heavens. The most remarkable of these 
are the grand reflector constructed by Lord Rosse, an Irish nobleman, 
and the great refractors belonging respectively to the Pulkova, the Cin- 
cinnati, and the Cambridge observatories. 

Lord Rosse 's telescope considerably exceeds in dimensions and in 
power the great 40 feet reflector of Sir William Herschel, being fifty 
feet in focal length and having a diameter of six feet, whereas that of 
the Herschelian telescope was only four feet. This unexampled mag- 
nitude makes this instrument superior to all others in light, and fits it 
pre-eminently for observations on the most remote and obscure celestial 
objects, as the faintest nebulae. But its unwieldy size, and its liability 
to loss of power by the tarnishing or temporary blurring of the great 
speculum, will render it far less available for actual research than the 
great refractors that come in competition with it. 

Until recently, it was thought impossible to form perfect achromatic 
object-glasses of more than about five inches diameter ; but they have 
been successively enlarged, until we can no longer set bounds to the 
dimensions which they may finally assume. The Pulkova telescope 
(at St. Petersburg) has a clear aperture of about fifteen inches, and a 
focal length of twenty-two feet. That of Cincinnati is somewhat small- 
er, its object-glass being twelve inches diameter, and its length seven- 
teen feet ; but in power it is supposed to be fully equal to the Russian 
instrument. The telescope recently acquired by Harvard University, 


is perhaps the finest refractor hitherto constructed. Its dimensions are 
nearly the same with those of the Pulkova instrument, but its perform- 
ances are thought to be superior even to that. 

These magnificent telescopes have afforded views of celestial objects, 
more splendid and exciting than any previously enjoyed by man. In a sci- 
entific point of view, the most interesting of these revelations consist in the 
resolution of nebulae before deemed irresolvable, and thus countenancing 
the idea that this term is applicable only to the comparative powers of 
our instruments; that, if any objects of this class remain unresolved, it 
will only be because the telescope has not yet acquired the requisite 
power to separate them into stars. Under these mighty instruments, 
what was before a faint wisp of fog on the confines of creation, expands 
suddenly into innumerable suns, composing a glorious firmament of 
stars. The Cambridge telescope has succeeded in the resolution of the 
great nebula of Orion more complete than had been effected even by 
Lord Rosse's " Leviathan" Reflector, and is thus proved to be one of the 
finest instruments (probably the finest refractor) in the world. 












JK3T THESE LECTURES are delivered to the Class after they have 
finished their recitations in the Text-book, and are therefore presumed to 
be acquainted with the Elements of Astronomy. To persons attending 
the course, who have not read this or some similar work, it is recom- 
mended to accompany the Lectures with the perusal of the author's 
" Letters on Astronomy." 

In the delivery of the Lectures, the pupils are supposed to have these 
Outlines before them, and to accompany the lecturer through the suc- 
cessive heads of each Lecture. They will afterwards form, the basis of 
the examinations, both public and private, on the same subjects. 






PLAN OF THE COURSE To consist chiefly of an exposition of 
the Structure of the Universe Prefaced with a concise view of 
the History of the Science Meaning of the phrase " Structure 
of the Universe," and how this subject differs from the elements 
of the science contained in the text-book here all things consid- 
ered in their relations to one grand whole, the System of the 
World. More interesting than the mere study of the elements, 
as the study of the classics is more interesting than that of gram- 
matical rules, which are only preparatory. Things usually more 
interesting in their relations, than in their individualities. 

Subjects of the present lecture, the different classes of astrono- 
mers Laws of reasoning in astronomy Pleasures and advan 
tages of the study of the heavenly bodies. 

separate nature of the gifts that have characterized different as- 
tronomers these to be considered in estimating the respective 
authority of each. 

1. The meditative class, as Pythagoras, Copernicus. 

2. The mechanical classes Tycho Brahe, Sir William Herschel. 
2. The mathematical class, as La Place, Le Verrier. 

4. The class of consummate astronomers those who have 
united these different qualities, as Kepler, Galileo, Newton. 

slight evidences or feeble analogies, when it is supposed full evi- 
dence is unattainable Unreasonableness of skepticism and incre- 
dulity in regard to the established truths of the science. Advan- 
tages of having in view a few leading principles of reasoning. 

1. The possibility of a thing but slight evidence of its reality. 

2. The same causes produce the same effects proper meaning 
and restrictions of this law. 


3. The argument from analogy explained its use and abuse 
in astronomy. 

4. The argument from authority to be received with caution, 
and, of each astronomer in his own peculiar sphere of excellence, 


1. To commerce and navigation. Stars earlier guides than the 
compass Now far more accurate Example of the accuracy of 
the lunar method given by Basil Hall Constancy of the motions 
of the heavenly bodies, and the perfection of instrumental meas- 
urement, as asserted by Dr. Bowditch. 

2. To chronology. Use of the precession of the equinoxes, of 
solar and lunar eclipses, of the situation of the pole of the earth 
among the stars for fixing remote dates. Command which the 
astronomer has over time, past and future. 

3. In furnishing standards of weights and measures. Impor- 
tance of this subject in business transactions Difficulty of finding 
perfect standards Standards derived from an arc of the meridian, 
and from the pendulum. 

4. Intellectual advantages. Fulfils the two great purposes of 
education to enlarge the mental powers, and to store the mind 
with important truths. Employs a variety of faculties stimulates 
to new efforts induces habits of profound reflection favorable 
to the development of both the imagination and the intellect. 

5. Moral advantages. What class of astronomers have been 
devout, and what class undevout men tendency of the contem- 
plation of the heavens to awaken devotional feelings Characters 
exhibited by great astronomers tendency to inspire devotion 
and the love of truth noticed by the ancients, as Lucretius and 
Cicero modesty of great astronomers asserted by Chalmers. 


1. Delight experienced in contemplating the starry heavens 
with all the lights of modern astronomy. 

2. Pleasure of recognising known constellations when in for- 
eign lands. 

3. Interesting reflections connected with the immutability of 
the heavens. 

4. Fascinating nature of the study, both to the mathematician, 
who discerns new laws, and to the practical astronomer, who 
discovers new worlds. 



Two elaborate histories of astronomy by Bailly and Delambre. 
Four ancient nations that cultivated this science antiquity of the 
study two leading objects, Eclipses and Astrology. 



Eclipses. Importance attached to them supposed connection 
with impending disasters how regarded among the Chinese. 

Astrology. Objects rank of astrologers pretensions of the 
art horoscope. 

Chinese and Indian Nations. High pretensions to antiquity 
Eclipses recorded from a very high antiquity Obliquity of the 
ecliptic determined Astronomy of the Brahmins. 

Chaldeans. Excellence of their observations Discovery of 
the Saros. 

Egyptians. Cardinal points indicated by the sides of the Pyr- 
amids astronomical paintings and inscriptions. 

Greeks. Long after the preceding nations three successive 
schools, at Miletus, Crotona, and Alexandria. How the sages of 
Greece acquired their knowledge how the modes of instruction 
in those ancient schools differed from modern methods Tholes 
doctrines of his school. 

Pythagoras. Extent and celebrity of his school its date 
Great truths foreshadowed to Pythagoras Singular opinions en- 
tertained by him respecting the music of the spheres. State of 
astronomy in his time. 

Alexandrian School. Date by whom established great as- 
tronomers connected with it Hipparchus greatest astronomer 
of antiquity when he flourished, and where his discoveries in- 
struments trigonometry, plane and spherical tables of the sun 
and moon eccentricity of the solar and lunar orbits precession 
of the equinoxes motion of the apsides backward motion of the 
moon's nodes exact length of the year, and obliquity of the 
ecliptic his mode of calculating eclipses catalogue of the 

Crystalline spheres of Eudoxus. 

Ptolemy. Greatest writer among the ancient astronomers 
Almagest Ptolemaic system of astronomy. 

Romans. The little attention they paid to astronomy. 

Arabians. The astronomers of the Middle Ages. 



COPERNICUS, 1473. State of the age Class to which he be- 
longed His system of the world How his merits exceed those 
of Pythagoras State of the science at that age His proofs de- 
fective His system compared with that of Eudoxus, and with 
that of Ptolemy. 

TYCHO BRAKE, 1546-1473. Seventy-three years after Coper- 
nicus Birth and education Class to which he belonged His 


observatory Number and value of his observations What in- 
struments he had, and what he was destitute of His removal to 
Prague his weaknesses. 

KEPLER, 1571-1546. Twenty-five years younger than Tycho 
Brahe Birth and education Mysterium Cosmographicum, its 
object and character Characteristics of his genius Alliance 
with Tycho. His Laws -the first discovered in astronomy their 
importance. His eccentricities. 

GALILEO, 1564-1546. Eighteen years younger than Tycho, 
seven years older than Kepler Early mechanical studies In- 
vention of the telescope discoveries with it confirmation of the 
Copernican system Disputes with the Aristotelians persecu- 
tions Character of his genius a " consummate" philosopher. 

BACON. Cotemporary with Galileo their respective merits 
compared in philosophy and astronomy. 




NEWTON, 1642-1564. Seventy-eight years later than Gal- 
ileo his rank among the human race Saying of Bailly 
Characteristics of his genius Three capital discoveries Eight 
great discoveries of the 17th century, 1. Pendulum. 2. Tele- 
scope. 3. Logarithms. 4. Micrometers. 5. Algebra applied 
to Geometry. 6. Kepler's Laws. 7. Calculus. 8. Universal 
Gravitation. Newton's inquiries into the philosophy of Light 
and Colors. Discovery of the Law of Gravitation Why this is 
esteemed the most important principle in physics What it has 
done for the cause of truth [accounts for the celestial motions 
weighs the sun and planets determines the exact figure of the 
earth, and of every body in the solar system accounts for all the 
irregularities in the motions of the heavenly bodies accounts for 
the tides, and determines their height at every place suggests 
new fields of observation anticipates the discoveries of the tele- 
scope, and corrects the most refined observations determines 
the stability of the solar system predicts the exact return of 
comets and reveals to us new planets.] Whether the doctrine 
of universal gravitation will ever be superseded. 

Personal qualities and virtues of Newton. 

OBSERVATORIES. Greenwich Astronomers royal Extent and 
accuracy of their observations importance of these. 


SOCIETIES. Royal Society of London Academy of Sciences 
at Paris Astronomical Society of London. 

Astronomers of England and France Respective excellence 
of each Other great astronomers of Europe State of astronomy 
in our own country. 



Remaining part of the course intended to explain the structure 
of the universe Embraces two inquiries, (1) What bodies com- 
pose the universe, (2) How they are arranged so as to form the 
system of the world. Order of subjects, the Sun, Moon, Planets, 
Comets, Stars, and Nebulae Mechanism of the Universe. 

SUN. Important relations to us as the source of light, heat, and 
attraction Consequences were the sun withdrawn. 

Diurnal Revolution. Phenomena at the equator, and in differ- 
ent latitudes Oppressiveness of his direct rays seems directly 
overhead throughout the torrid zone Opinion of the ancients that 
the torrid zone was uninhabitable Provisions of nature for miti- 
gating the intensity of heat. Phenomena of different latitudes 
from the equator to the pole at mid-summer at the equinoxes 
at mid-winter Appearances of the sun at these different sta- 
tions at different seasons of the year Phenomena of twilight in 
various countries lasts all night at midsummer beyond 48 1 of 
latitude length of twilight in our latitude length of the shortest 
night ditto of the longest night. In travelling round the earth 
eastward, a day gained westward, a day lost Case of two trav- 
ellers going round the earth, and returning to the same place 
also, when meeting at some other point, as at the Sandwich Islands 
Case of ships, and of missionaries Effect of these principles on 
the Sabbath When the Sabbath begins Case where the twilight 
lasts all night where it is continual day where the parties meet 
on opposite sides of the earth. When is the design of the ordi- 
nance fulfilled ? Reasons for consulting expediency. 



ANNUAL REVOLUTION. Mode of distinguishing between the 
diurnal and annual motions of the sun cause of each explained 
Compatibility of the two motions Solar days longer in winter 
than in summer Sun eight days longer on the north than south 


of the equator why Greater nearness in winter compensated by 
the shorter time, so as to preserve an equality in the distribution 
of heat. 

Obliquity of the Ecliptic. Modes of finding it practised in 
ancient and in modern times Its amount recorded at different 
epochs Its annual variations Present amount. 

ROTATION OF THE SUN. Period Telescopic appearance of 
the sun. 

SPOTS. Description extent cause. 

Galileo's hypothesis. Fiery sea Volcanic scoriae objections. 

Lalande's. Recession of the fiery sea Solar mountains and 
valleys objections. 

HerscheFs views of the nature and constitution of the sun 
Supposed atmospheres whence the light and heat false reason- 
ing whether the surface of the sun is in a state of combustion or 
of ignition. 



Its uses for light, for tides, for months, and for longitudes 
comparative nearness Distances of the other heavenly bodies 
measured by millions, of the moon by thousands Appearances 
to the naked eye of its phases, and of its light and shade, when full. 

REVOLUTION AROUND THE EARTH. Changes in its path, its ve- 
locity, its nodes Numerous irregularities their general cause 
why so much more numerous in the moon than in any other of 
the heavenly bodies Pains taken to perfect the lunar tables 
different means employed for this purpose. 

ing the moon Magnifying powers employed for different pur- 
poses Successive appearances at different ages Best view at 
quadrature. Objects to be particularly noted. 

1. Broken surface extreme irregularity. 

2. Appearance of the terminator Single mountains described 
Height of the lunar mountains. 

3. Circular chains long ridges. 

4. Radiations from Tycho, Kepler, and Copernicus. 

5. Valleys and craters resemblance of these to volcanic cra- 
ters on the earth mode of formation. 



None seen in eclipses Supposed appearance of twilight No 
change of place by refraction in a star undergoing occupation 
Atmosphere, if any, very small and rare. 

not such no clouds nor fogs no atmosphere of watery vapor. 

VOLCANOES. Proofs of their existence objections considered. 

LUNAR INFLUENCES. Supposed influence on the weather on 
diseases on vegetables. 

Whether we can ever hope to see lunar inhabitants or their 
works ? Limits to the powers of the telescope. Effect of apply- 
ing different magnifiers from 240 to 10,000. 



In nearness, next to the sun and moon the sun, moon, and 
planets, compose one family in the stellar universe. Influence of 
the planets upon our world slight, though considered in the days 
of astrology as very great. 

NUMBER. How many were known to the ancients Analogy 
traced by Kepler to the five regular solids Whole number, in- 
cluding the earth and moon, seven Supposed by the Aristotelians 
to be necessarily seven Reasoning of the Roman doctor against 
Galileo's discovery of Jupiter's satellites. Uranus added in 1781 
Ceres, Pallas, Juno, and Vesta, about the commencement of the 
present century more recently, Astra3a, Hebe, Iris, and Neptune. 

Remarkable discovery of Neptune by Le Verrier his reasons 
for supposing its existence Mode of investigation Elements 
determined previous to observation Discovery with the tele- 
scope Similar results obtained by Adams Respective merits 
of these astronomers How their results have been modified by 
Walker and Pierce This planet seen by Lalande in 1795, and 
recorded as a fixed star Traced back to this place by Walker. 

Great distance of Neptune from the sun Long period of rev- 
olution Magnitude, density, and orbit eccentricity. 

Appearances of the different planets to a spectator on the sun, 
in regard to size, light, velocity, &c. 




Distinction between laws and individual facts No laws in as- 
tronomy known to the ancients the first those discovered by 
Kepler Our knowledge advanced rapidly by laws, slowly by 
individual facts. Laws reveal systems and comprehensive de- 

1. KEPLER'S LAWS. History of their discovery Known to 
Kepler as matters of fact, and not of demonstration First dem- 
onstrated by Newton. 

(1) Figure of the orbits how these differ from the eccentric 
orbits of the ancient astronomers. 

(2) Equality of Areas described by the radius-vector Great 
utility of this law in calculating elements. 

(3) Distances and periodic times. Value of this discovery 
Exultation of Kepler on finding it Reduces several of the 
most important particulars of the planets to exact numerical 


Kepler's third law and the doctrine of universal gravitation 
Velocities, distances, periodic times, and gravitation towards the 
sun, all adjusted to one another, like the hour, minute, and second 
hands of a clock Relation between mass and velocity Velocity, 
distance, time, and force of gravity, compose a series in geomet- 
rical progression of which the first term is the ratio How derived 
from Kepler's third law and the law of gravitation. Any one of 
these terms being given, the rest may be found illustrations 
Method of finding the masses and densities of the planets ex- 
plained and illustrated. 



1. GREAT MECHANICAL LAWS which belong to every system of 
bodies, connected together by mutual relations, but perfectly veri- 
fied only in the heavenly bodies. 

(1) Conservation of areas. 

(2) Conservation of the center of gravity. 

(3) Principle of Least Action. 




Experiment with a suspended ball Effect of gravity alone- 
of the impulsive force atone Gravity accounts only for the con- 
stant deviation from a straight line A primitive impulse indispen- 
sable No cause for it MOW existing Must have been applied at 
some previous period of its existence, and then withdrawn Argu- 
ment for a First Cause Evasion of this argument how obviated. 

Why a planet or a comet turns about at its aphelion, and what 
keeps it off from the sun at its perihelion. 


(1) Of the bodies composing it. 

(2) Of their relations to one another. 

Resemblance of the subordinate systems to the whole Uni- 
formity of plan throughout General motions from west to east 
Exception to this uniformity in the satellites of Uranus. 

Constancy of the law of gravitation. 


" Problem of the three bodies" Causes of disturbance existing 
in the system Stability of the system first determined by La 
Grange, and confirmed by La Place Its mathematical expression 
[The mathematical formula which expresses the effect of all the 
perturbations, is contained in the sines and cosines of arcs, which 
necessarily vary between zero and radius] the perturbations 
oscillate about a mean value Grand axes always constant also 
the periodic times. 

Doctrine stated. 

1. That the earth revolves on its own axis. More simple 
Analogy Spheroidal figure Diminished weight at the equator 
Bodies fall eastward of their base. 

2. That the planets, including the earth, revolve about the sun. 
Phases of Mercury and Venus Greater symmetry of the 
movements and orbits of the superior planets when referred to 
the sun as a center. 

3. That the motion of the earth around the sun is especially 
indicated by analogy by Kepler's law by the retrograde mo- 
tions of the superior planets by the phenomena of meteors. 



Circumstances unfavorable to the doctrine Extremes of neat 
and cold of light of gravity Want of atmospheres Such 


beings as inhabit this earth could not inhabit the planets, but still 
some reasons for thinking that they have their appropriate inhab- 

1 . Reasons for believing that the planets were designed for the 
same purposes as the earth. 

(1) From the uniformity of plan manifested throughout nature. 

(2) From the fact that the earth is a member of a system, and 
not the leading member. 

(3) From the consideration that all things have their use illus- 
trations of this point Uses of the planets to our earth very slight. 

In order to learn the purpose for which the planets were made, 
we inquire, 

2. For what purposes the earth was made. 

(1) Animal life. Its multiplication in every part of the earth 
Great range from the whale and the elephant to the infusoria- 
Multiplication of life even in the polar seas. 

(2) Not animal life in general, but Man the great purpose for 
which the world was made. All things for his use. 

First, the powers of nature. 
Secondly, the productions of nature. 
Thirdly, the beauty and sublimity of nature. 
Fourthly, the power of exalting nature of developing its 
powers of compounding and forming new creations. 

(3) Argument from analogy applied to the planets to prove 
that life is their general, and intelligent beings their special 

(4) Direct evidences that the planets are inhabited. 



1. HISTORICAL NOTICES. Pythagoras Aristotle Tycho 
Newton Circumstances that render them intrinsically interesting 
History of Halley's Comet. 

2. LAWS OF THEIR MOTIONS. In what respects they are alike, 
and in what unlike those of the planets. 

difficulty Method of finding the elements These may serve to 
identify the comet with one that has appeared before, but do not 
give the periodic time How the time is ascertained Exemplified 
in Halley's comet. The approximate time being found, to find 
the exact time Number and sources of the perturbations Labo- 
rious nature of the calculations Exemplified in the returns of 
Halley's comet in 1759 by Lalande, and in 1835 by Pontecoula^ 


How far the predictions of astronomers, on the last return, were 

4. PHYSICAL NATURE OF COMETS. Constitution of the three 
several parts Extent and tenuity of the train Exceedingly small 
mass Whether so rare a body would obey the laws of gravita- 
tion and projection Whether the matter is self-luminous Hy- 
potheses to account for the formation of the tail by the impulse 
of the sun's rays by the action of heat Difficulties attending all 
hypotheses yet proposed. 


(1) Dangers in various parts of nature, as from extremes of 
heat and cold from the perturbations of the moon and planets 
Guards set to prevent mischief Whether any guards can be de- 
tected in the case of comets. Two especially, in the swiftness 
of their motions, and the smallness of their masses also from the 
fact that they are subject to laws. 

(2) Consequences were they to strike the earth Threatening 
circumstances attending the great comet of 1843. 

(3) Question of a resisting medium, and its final effect upon 
the motions of comets. 



Solar system the chief object of attention to astronomers before 
Sir William Herschel His forty-feet telescope Rosse's fifty-feet 
reflector Great refractors recently constructed, as the Pulkova, 
the Cincinnati, the Cambridge instruments, respectively Com- 
parative advantages and disadvantages of reflectors and refractors 
peculiar advantages of large telescopes. 

Fixed stars. Why so called Immutability of the constella- 
tions, contrasted with the mutability of terrestrial nature. 

Twinkling of the stars explained. 

Catalogues of the stars by Hipparchus, Tycho, Flamsteed, 
Lalande Astronomical Society Bessel's zones Berlin charts. 

DISTANCES. Eight of the greatest discoveries in astronomy : 
1. Kepler's Laws. 2. Law of Universal Gravitation. 3. 
Weighing the Sun and Planets. 4. Demonstrating the Stability 
of the Solar System. 5. Measuring the Velocity of Light. 6. 
Exact Determination of the Periodic Time of a Comet, as Hal- 
ley's. 7. Discovery of the Planet Neptune. 8. Measuring the 
Distance of a Fixed Star. Proofs of the immense distance of 


the stars, from the want of parallax from the effect of the largest 

Researches for an annual parallax by Flamsteed, Bradley, 
Brinkley, Bessel Observations of Bessel on 61 Cygni Amount 
of its parallax Whether it may not be fallacious Illustrations 
of the distance of 61 Cygni by the time occupied by light bv a 

Supposed parallax of Alpha Centauri of Sirius. 




(1) Double, triple, and multiple stars. Great number of such 
groups Sometimes merely optically double Revolutions of the 
binary stars Proof which they afford of the extent of the law 
of gravitation. 

(2) Clusters. Pleiades Beehive in the head of Orion in 
the sword-handle of Perseus. 

(3) Nebula. Description Various forms Extent Resolva- 
ble and irresolvable Whether this distinction is absolute or rel- 
ative Bearing of recent discoveries of the great telescopes on 
this question Researches of Mason and Smith on nebulae Ob- 
servations of Sir John Herschel in the southern hemisphere. 

(4) Galaxy. Its figure Constitution according to Sir William 
Herschel The relation of our solar system to it How our posi- 
tion in it is determined. 


(1) Material bodies obey the law of attraction. 

(2) Larger than our earth Sum of the masses of both the stars 
in 61 Cygni half that of the sun Orbit twice that of Uranus 
Period 540 years Sirius equal to two suns, or, according to Wol- 
laston's experiments upon its intrinsic light, to fourteen suns. 

(3) Light of all has the same velocity Velocity uniform 
Constant of aberration the same in all. 

(4) Light of the stars not polarized, therefore direct, and not 

3. FINAL PURPOSE of the stars. 

Not for ornament not to give light by night not for naviga- 
tion Are suns of other systems Uniformity of plan in the works 
of creation Purpose of our sun for light, heat, and attraction to 
planetary worlds Life the great object of these All for intelli- 
gent, immortal beings. 




L Propriety of reasoning from the uniformity of plan in nature 
Deducing what we cannot see from what we do see. 

2. What we actually see. 

(1) Subordinate systems, as Jupiter and his satellites. 

(2) The solar system. 

(3) The binary stars. 

(4) Higher systems in groups, clusters, and nebulae. 

3. Probable prevalence of the law of universal gravitation 
through the universe. Consequences of such a law Analogy 
leads us to expect revolutions Mere attraction without it tends 
to ruin Proper motions detected among the stars These mo- 
tions of revolution. 

4. Grand machine of the universe. Probability that it is con- 
structed after the model of the solar system Its actual organi- 




CENTER OF THE UNIVERSE (?) Attempts of Maedlerto find the 
center about which the stars of our firmament revolve Fixed in 
Alcyone, the principal star of the Pleiades Reasons for selecting 
this point Whether any thing more is necessary than a common 
center of gravity Movement of the solar system, and of various 
other stellar systems in respect to this point. 


1. Herschel's views of the progressive state of the nebulae 
Whether changes now take place within short periods Mason's 
researches on this subject. 

2. La Place's hypothesis of the formation of the solar system 
Principal object to account for the revolutions all in the same 
direction Principles of it stated Supposed illustration by ex- 
periment Phenomena explained by the hypothesis Comets not 
included Objections considered whether the doctrine is neces- 
sarily atheistical whether such a mode of formation is possible 
whether probable Backward motion of the satellites of Uranus 
Bearing on this hypothesis of late telescopic discoveries among 
ths nebulas. 



1. Relations of Astronomy to Natural Theology. 

2. Place which our world and which man holds in the universe 
Supposition that both are too insignificant to be objects of high 
and peculiar interest to the Creator, considered 

3. View which the Creator took of his own works when be 
pronounced them " very good." 




1. Castor. 2. 7 Leonis. 3. 39 Drac. 4. X Oph. 5. 11 Monoc. G Cancri. 

Revolutions of y Virginis. 

1837. 1838. 1839. 1840. 1845. 1850. 1860. Orbit. 



I have examined " Coffin's Conic Sections and Analytical Geometry," and hesitate not 
to say, that its size, plan and general arrangement, ought to procure for it a place in the 
hands of every teacher who wishes to give a liberal knowledge of Mathematics, without 
abridging the other branches of a good education. I am happy to say that your book is 
better adapted to my present wants than any I have seen. I have therefore concluded 
to adopt it for my next class. 

By bringing out this work you have advanced the cause of science, merited the thanks 

)f the lovers of learning, and done a lasting good to the public As a general principle, 

f am opposed to the shortening of books in order to shorten the course of study. The spirit 
)f the day is to get along most rapidly, not most thoroughly ; by bestowing the least pos- 
sible labor, both in physical and mathematical science. But from the examination I have 
been able to make of your work, I find it contains some important matter, not found in 
other works of a like character, whilst nothing that is valuable is omitted. The compass is 
contracted, but the whole circle is there. It is just the work we need, and I shall adopt it 

in our Academy in preference to other treatises upon these subjects Certainly by 

your combining the geometrical and analytical methods, you present the subject more 

clearly Your demonstrations are admirable ; they are so concise, yet so simple, that 

no student can pass over them without fully comprehending them. 


I have received and read with great pleasure your " Elements of Conic Sections and 
Analytical Geometry," and cannot hesitate to commend it as an excellent system. It 
embraces in a parallel view, the Geometrical Theory and the Analytical investigation of 
the properties of the Conic Sections the method which I have always pursued in the 
instruction in my department, but have felt the want of a proper text-book on the subject 
This want has been met by your Treatise, and I have adopted it as a text-book with 


Your work was examined, I do not say by me, but by judges more competent than I 
am. We are unanimous in saying that your demonstrations, with respect to perspicuity, 
are to be highly praised. You are clear, concise, and, whilst you avoid obscurity, you 
find the means of being both exact and brief. This is no small merit, and you are enti- 
tled to the gratitude of students, to whom you spare useless difficulties and exertions. 


Your treatise on " Analytical Geometry" Appears to me to possess advantages over 
any other treatise which I have examined. vOne of these advantages I consider to be, 
the division into Propositions distinctly enunciated, so as to keep constantly before the 
mind of the student the object of his investigation. Another advantage consists in the 
introduction of numerical examples, which afford a useful exercise to the student, and 
present the best test of the clearness of his conceptions. 


I have received and examined a copy of your recent work on " Conic Sections." .... 
The manner in which you have introduced both the geometrical and analytical demon- 
strations, ought to secure success to your work. You present what I think is needed by 
the American student. In the selection of problems and in the elucidation of parts that 
often prove unreasonably obscure to beginners, you have been very happy I recom- 
mend the work with confidence to the attention of Instructors and Students. 



I sent to New Orleans for a copy of your treatise on the " Conic Sections and Analyt- 
ical Geometry," which I have received and have examined with some care and much 
satisfaction. I think after the perusal I have given it, I may safeJy pronounce your trea- 
tise eminently suited for use in colleges ; it is sufficiently elaborate, the arrangement i& 
good, and there is a plainness and simplicity about it which I admire very much. 

MESSRS. COLLINS & BROTHER. Gent. : Illness has prevented me from examining 
Prof. Coffin's " Treatise on Conic Sections" till very recently. I am greatly pleased with 
the work, and think that the author has succeeded in developing the most interesting and 
important properties of these curves with unusual simplicity and brevity in the demon- 
strations. To this, the common property of the curves, assumed as their fundamental 
characteristic, is, by the skill and tact of the author, made to contribute very much. 
This property, while it unites the three curves in a common bond, and gives them a 
common source, is that on which the more useful properties seem most immediately to 
depend, and from which they are most readily and naturally deduced. In the Geometri- 
cal portion there is a very just medium preserved, in the extent to which the discussion 
of the curves is carried, the investigations being limited to the most essential and practi- 
cally useful properties, and requiring no useless expenditure of time on points merely spec- 
ulative or curious. The second part, devoted to " Analytical Geometry," is a very im- 
portant addition to the work, and one, I think, without which the student will be very 
illy prepared to make his knowledge of these curves, readily and extensively available in 
their application to astronomical investigations. The work I regard as. a very decided 
improvement on the old systems and treatises, and think in its preparation and publication 
a very valuable contribution has been made to educational instrumentalities. 

Very respectfully yours, 



" Elements of the Conic Sections and Analytical Geometry, by JAMES H. COFFIN, A. M. 

Professor of Mathematics, $-c., in Lafayette College" 

In this treatise the doctrine of the Conic Sections is taught first geometrically ; and in 
a second part, the student is taught how to represent lines, curves, and surfaces analyti- 
cally, and to solve problems relating to them. We have examined the work with care, 
and testify to the skill, tact, and neatness of its expositions. Most books of Analytical 
Geometry are blind to scholars ; most of them never learn, unless they have a teacher 
unusually skilful and diligent, how to interpret algebraical expressions, or how to make 
practical use of equations. It is precisely for its clearness, its practical character, and its 
adaptation to the work of the recitation room, that we heartily commend this volume. 


I have recently had an opportunity of examining your treatise on " Conic Sections and 
Analytical Geometry," and am happy to state that, in my opinion, it is a work of much 
merit. The selection and arrangement of the Propositions appear to me judicious ; and 
the definition which you adopt, at the commencement, has certainly the advantage of 
supplying more simple and obvious demonstrations, in place of those whose prolixity and 
abstruseness have hitherto rendered this very important portion of Mathematics so for- 
midable to learners. The first part appears to contain all the properties of the Sections 
which are required in the branches of Mathematical Philosophy usually embraced in our 
College courses ; and the second part furnishes a valuable introduction to Analytical Ge- 
ometry, rendered more valuable by its immediate connection with what precedes, and by 
the practical nature of the problems proposed for investigation by analysis. I intend in- 
troducing it as a text-book in this Institution. 

I think it decidedly the best work in print upon the subject. 



Invites the attention of those interested to the following valuable books, which, it 
is bt'lievi'd, will be found on examination unsurpassed as text books. 


I. Primary Arithmetic. An introduction to Adams's New Arithmetic, revised edition. 

II. Adam*'* New Arithmetic, revised edition; being a revision of Adams's New Arith- 
metic, lirst published in 1^7. liino. halt bound. 

A Key to the above is published separately. 

III. Mensuration, Mechanical Towers, and Machinery, a sequel to the Arithmetic. 

IV. Bookkeeping by Single Entry, accompanied with blank books for the use of learners. 

Abbott's Arithmetic. 

The II on nt Vernon Arithmetic. Part I., Elementary. By Jacob and Charles E. Abbott 
The Mount Vernon Arithmetic. Part II., Vulgar and Decimal Fractions. By Jacob Ab- 
bott. 12mo., half bound. 

McCnrdy's Geometry. 

First Lessons in Geometry. By D. McCurdy. 

Charts to accompany the "First Lessons," on rollers, size 34 by 48 inches. 
Euclid's Elements, or Second Lessons in Geometry. By D. McCurdy. 12mo. half bound. 

Preston's Bookkeeping. 
District School Bookkeeping. Quarto, stitched. An excellent work for beginners ; printed 

on handsome demy paper for practice. 
Single Entry Bookkeeping. 8vo. cloth sides. 
Bookkeeping, by single and double entry. 

These three excellent works are by Lyman Preston, the author of the Interest Tables, &c. 

Coffin's Conic Sections. 

Elements of Conic Sections and Analytical Geometry. By James H. Coffin, Prof. Mathe- 
matics, Lafayette College, Pa. 

Day's Mathematics. 

I Course of Mathematics, containing the principles of Plane Trigonometry, Mensuration, 
Navigation, and Surveying. By Jeremiah Day, D. D. LL. D., President of Yale Col. 8vo. sheep. 


<) I nisi <:'* Rudiments of Natural Philosophy. 18mo. 
Rudiments of Astronomy. 18mo. 

These two works are intended to be used in district and common schools, where merely the 
elements of the sciences are taught, and also as preparatory class-hooks for the younger scholars 
in academies; for these purposes they are admirably adapted. They are in a neat 18mo. form, 
well bound in cloth, with leather back, and are illustrated by numerous engravings. 
Olmsted'ii School Astronomy. 12mo. 

A compendium of Astronomy for the use of pupils in the higher classes of common schools 
and in academies. 

Olmstcd's College Astronomy. 8vo., sheep. 
Olmstcd's College Philosophy, hvo., sheep. 

These works were prepared by Prof. Olmsted for the use of bis classes in Yale College, and 
are acknowledged to be the best works on the subject yet before the public; the author's name 
is, in itself, a sufficient guarantee of their excellence. 
Mason's Practical Astronomy. 8vo. cloth, leather back. 

This work, by the late F. P. Mason, is regarded by competent judges as the best work extant 
on the subject of Practical Astronomy. 

Solar and Lunar Eclipses, iamiiiarly illustrated and explained, with the method of calcu- 
lating them, as taught in the New England Colleges. By James H. Coffin, A. M. 8vo. 


The Mount Vernon Reader, in three numbers. By the Brothers Abbott 
Addick's Elements of the French Language. 12mo., half cloth. 
Girard's Elements of the Spanish Language. 

Two excellent rudimentary works. 

Badlnns's Common School Writing Books. A series of five numbers. 
The Governmental Instructor. A brief and comprehensive view of the government of the 

rnitt-d States and of the state governments, for the use of schools. By J. B Shurtleff. 
Kirkhnin's English Grammar. This work is more extensively used and approved than 

any work of its .size in tin; country. 

Whelpley's Compend of History, revised, with additional matter, by Joseph Emerson. 
John Rubens Smith's Drawing Book. Folio. 

The Oxford Drawing Book. 100 lithographic plates, with directions, &c. Quarto. 
TEsop's Fables. The most splendid edition ever published, with illustrations on wood, by 

Orr. i lowland, and Felter, after designs by John Tenniel. Crown 8vo., fancy cloth. 
Gnbriel, a story of Wichnor Wood. By Mary Howitt. The best taleof this popular authoress. 
Our Cousins in Ohio, a picture of domestic life in the West. By Mary Howitt. 18mo., cloth gilt. 
The Imitation of Christ. Hy T. a Kempis. Payne's translation, with essay by Dr. Chal- 

mers. The cheapest edition of this valuable work ever issued from the press. 
Abercrombie's Intellectual Powers. Inquiries concerning the Intellectual Powers, and 

In ve-tigatum of Truth. By John Abercrombie. 12mo., cloth. 
Abercrombie's Moral Philosophy. 

These two works of Abercroinbie'si are made much more valuable by the additions and ques- 

tions to I his edition by Jacob Abbott. 
D>moud's Essays on the Principles of Morality, and the private and political rights and obliga- 

tions of Mankind. This is regarded as the best work on Ethics in the language. 12rao. 

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